As with all such scenarios the exact geopolitics of how the USA comes to acquire the north and exactly how much of it where, and specifically when, matters tremendously.
I interpret the spirit of the OP question to be "how much did Britain retaining the northern lands of OTL influence demographics?" And therefore, I think we need the cleanest sheet possible. The American Revolutionary War managing to secure 100 percent of all British territory north of Florida is something I've engaged with before. It is very much a long shot, and to stick probably requires the French and allies to be more successful than they were OTL, so as to put Britain on the ropes more stringently--and yet such a strong French position raises the possibility of the King of France demanding New France or parts of it back, which partially defeats the purpose.
I'm going back to 1775 or so then, and perhaps we need some earlier POD seeds planted to make this work. In bare bones, the US Patriot offensive to take Quebec and Montreal is more successful, followed (perhaps with the help of respected royal French advisors not unsympathetic to the republican/Enlightenment aspects of the ideology of the AR) by successful diplomacy in winning over at least grudging acquiescence of the Francophone powers that be in "Quebec." It might be more logical for a subsequent Francophone US state to be called "Canada" or "Nouveau France" but to avoid confusion l'll settle on Quebec. State of. The French king decides not to press to have it restored to his crown and instead banks on gaining American support with magnanimity on the point. Meanwhile the Patriot cause in Nova Scotia is somewhat stronger, again perhaps French diplomacy papers over rivalries with Massachusetts and smooths the way. The British however do hold strong in Halifax itself, as they do in Newfoundland and Labrador and Rupert's Land mainly because hardly anyone can get there. But holding the NS countryside is tougher; sometimes the Regulars and Royalist auxiliaries surge out and seize most of the peninsula, but then they lose parts of it to insurgency. The general sweeping of war up and down NS drives many settlers there to relocate to the west, in OTL SE New Brunswick beyond the narrow neck, which the British seize and hold with expeditionary force that drains their ability to rule securely elsewhere. Of course as OTL a big part of the population is Loyalist/Tory anyway, and the Patriots are reduced to some limited holdings losing population to the refugee settlements to the west.
Also, having lost access to Quebec with Yankee forces including increasingly US loyal Quebecois militia, advancing down the St Lawrence shores on both sides and mustering a bit of naval opposition to the RN in that river's gulf, the British dig in hard on Long Island, and refugee Loyalists take refuge there. Here I refer to the whole complex of islands including Manhattan east of the US continent. Of course British forces often hold much of the mainland opposite too but insecurely, and Loyalist refugees don't feel safe until they are on the Islands, which are largely purged of Patriots and sympathizers.
Eventually, with the French being somewhat more successful in the other theaters of this inter-power war, when the British come to terms, France is able to get the British to surrender all claims on the US continent--except Nova Scotia and the Hudson mouth island complex. In the case of NS there is some bad blood among the Patriots there since they held some strongholds, whereas the treaty concedes the entire peninsula, pretty much exactly the OTL province boundaries. But to compensate, the vast state of Arcadia is created--much of OTL Maine (the far north and a portion east of a river that OTL was the boundary between Massachusetts colonial ventures and those out of NS) and all of OTL New Brunswick and St John's island, which is OTL renamed Prince Edward Island but obviously would not be so named here. The portion of NS population that survived the war as Patriots is largely relocated to the southeast of this zone anyway, and the rest join them, leaving NS entirely under Loyalist control and with few covert Patriots, most of whom either change their minds or emigrate to Arcadia or some other US state eventually.
Arcadia is also a semi-concession to the Quebecois and Francophone Americans. The exiled "Acadians" are invited back, and the northern part of Arcadia (spelled Acadie in French) is informally expected to be an expansion zone for Francophones, as it turns out not quite as many as expected and mostly from Quebec. St John's Island is somewhat fortified and becomes a USN base, which causes it to attract a fair amount of Anglo settlers, but under the accords with France Francophones have at least grudging respect in Arcadia. The "Anglo" center of Arcadia in early days is the shore opposite Nova Scotia south of the peninsula; Arcadian militia (including rising numbers of Francophones, who all speak some English more or less) and US Army forces hold forts on the US side versus British forts on the British side dividing the narrow isthmus.
Quebec, as a state, is largely under conservative Francophone control, heavily influenced by the Catholic hierarchy, but American republicanism guarantees the Anglophone minority and religious or political dissidents some rights. US successes are gradually winning over the Quebecois masses to US loyalty, and gradually Quebec is secularizing. It is not Constitutionally possible (nor would it be under the Articles of Confederatio before 1788) for Quebec to forbid Anglo-American immigration, but various gentleman's agreements tend to discourage it, except for persons keen to more or less Frenchify. The Quebec state government aggressively promotes French culture; at least two major universities, one nominally secular under the Quebec state government and one founded and run by Catholic Church officials, emerge as the Francophone equivalent of the Anglophone Yankee Ivy League schools. (Educated Quebecois, and most common folk, do learn a lot of English, the more ambitious ones are quite proficient, but the emphasis is on pride in a Franco-American identity; their English has an accent they take pride in as cultivated). Over time, Quebec and later Louisiana will draw in a lot of Francophone emigrants from Europe, and elements of French political and cultural factions will be mirrored more or less in parts of both; in Montreal, Quebec City and New Orleans in particular, some elements--some born in Europe, some very wealthy, some out of pretension--will mimic Parisian upper class French very closely. Others will evolve more distinctly Quebecois and Louisianien turns of speech.
In addition to an informal co-dominium in "Arcadie," the territories of OTL "Upper Canada" are vaguely penciled in for their expansion.
Meanwhile the British, cornered into losing most of their Quebec province (and losing Florida militarily to Spain, rather than merely conceding it postwar as OTL) dig in stubbornly on Rupert's Land, Labrador, Newfoundland and Nova Scotia and are conceded these, along with the greater Long Island complex. Perhaps the upshot of this would be for New York state to change its name, or else the British might not refer to the growing urban complex at the southwest end of the islands as "New York" but give it some other name, maybe say Manhattan in Long Islands. There is thus no NYC as such. In another thread on this very subject I offered the view that for about 100 years, the net growth of urbanization in the OTL NYC greater metropolitan area would be somewhat inhibited, as each nation's armies and naval forces glare at the other's across the dividing strait. To an extent this very demand for fortification will anchor a certain degree of development, as will the simple fact that New York Harbor has important advantages. Against this though US policy won't want to be too dependent on an urban area under British control, nor center their major naval harbors and construction yards at such pointblank range. The British will develop the northeast end of LI somewhat more as a reserve bastion in case of major and successful Yankee invasion.
In between, a lot of Loyalists from all 13 OTL US "colonies" in rebellion will settle most intensively on the Long Islands, to a lesser intensity in less hospitable (but more secure) Nova Scotia, and hardly at all in Newfoundland and Rupert's Land. If we unbutterfly enough to have a version of the OTL War of 1812 (I guess later, based on Napoleon Bonaparte lasting longer) the upshot is a shadowy low-troop number war out of northern Quebec and territories to the west against weak dispersed British/Hudson Bay Company forces with lots of Native allies, British landing attempts on the eastward and loosely held Quebec and Arcadian shores of the Gulf of St Lawrence, and intensive fighting on the narrow Arcadia/Nova Scotia isthmus border and between Long Islands and the various US shores facing it. Overall, I would see the war coming to the same sort of inconclusive status quo ante, perhaps taking longer however for the two nations to agree to gradually disarm their borders.
By the 1880s, a cusp is reached whereby the leadership of neither nation is at all likely to approve hostility against the other. By this time, the Long Islanders and Nova Scotians (incorporating Newfoundland and Labrador and perhaps Rupert's land) will have achieved representation in the Westminster House of Commons, and some American Lords would have been long before named. Thus these British holdings eventually become politically speaking integral to the UK. The political deal is not actually that great in the mid-19th century but gradually improves with telegraphy, telephony across the Atlantic, and faster travel across the ocean making it more and more practical for MPs of American ridings and boroughs to maintain constituency contact while still serving in London. As tensions ease between USA and UK, the points of contact go from being depressed no-man's land to fostering considerable economic growth--this is mainly true of the OTL NYC area, but note the region is coming from well behind OTL. On both US and UK soil it will grow and sprawl, but not to the same sort of megapolis as OTL.
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So what would this ATL setting imply for the demographics of the Greater USA including most of OTL Canada from the get-go?
First I want to emphasize what I think the knock on ATL effect of accepting Quebec as a nominally equal state while it remains overwhelmingly Francophone and initially largely under the thumb of conservative and chauvinistic French acculturation, initially with the Catholic Church hierarchy largely pulling the strings. In the most simple and one dimensional sense, overall I expect American anti-Papism to be somewhat moderated. In a more nuanced sense--to an extent it is actually intensified among a more alarmed radical minority who cry havoc. But between the direct influence of Catholic US citizens and numbers of WASP citizens who have more or less civil and profitable relations with their Catholic compatriots, the balance of opinion among the less frantic shifts a little bit the other way. In part this is because OTL US Catholics tended, again in misleading aggregate averages, to lean more liberal than the ultramontane positions of the 19th century Catholic Curia in Rome. This carries over, slowly at first but accelerating, to Quebecois and Louisianien Francophone populations. Relatively few of these (versus the pattern among American Catholic immigrants assimilating to anti-Papist US society) will convert to some kind of Protestantism, though quite a few will--but as self-perceived loyal Catholics many will take stands more liberal than the Curia likes, and others are likely to become atheist or agnostic. Contact with France and Belgium via trade and immigration from there is likely send waves of emigrants on the losing side of European politics.
Demographically France is a weaker source of immigration than usual in Europe, due to a remarkable decline in birthrate in the 19th Century. We might want to handwave that away but it would be special pleading I think. However, OTL a fair number of US immigrants were French or from other Francophone territories. We can expect a modest increase in this net movement, and a fair portion of these settling in either Quebec or Louisiana, to reduce the degree to which these regions were Anglicized OTL. As noted I think this general immigration will include exilic elites and radicals, and these will feed the "petit Paris" status of Montreal and New Orleans.
Meanwhile, as generations pass in which both Quebec and Louisiana prove loyal and strong US states, a general American sympathy for things French in America will be deepened and strengthened. French as a second language, mastery of which is taken as a passport to pretensions of higher culture in the USA, will be wider and deeper, reinforced by grassroots encounters between Quebec and Louisiana settlers in the West in general and migrants to the US major cities. Anglo-Americans might choose to go to the Francophone universities for the cultural cachet.
I would hope this baked in early semi-bilingualism would lead to trilingualism--between lowering the resistance to Catholic citizenship, and the minority self-advocacy of Francophones, there might be easier and earlier acceptance of Spanish speaking people if we assume the USA conquers pretty much as OTL. OTL Florida had few Spanish speakers in it to be sure, but perhaps New Mexico for instance does not have to wait until the 1910s as OTL to become a state. Perhaps Puerto Rico would achieve statehood fairly early and maybe the USA would annex Cuba and make it a state fairly promptly as well.
This might be associated with somewhat more and earlier emigration from Latin America.
Overall however, there is little reason to think US immigration would be much more than OTL. A little more from France, a little more from Latin America, but pretty much as OTL.
Meanwhile, would the emigrants from Britain and elsewhere (largely British possessions, outside of Europe) who went to Canada OTL go to the USA instead?
Well, quite a few clearly would not. On one hand, loyalist Britons would definitely not. There won't be a lot of room in Nova Scotia or The Long Islands, once the OTL Loyalists (and then some) who settled largely in Upper Canada have taken refuge there instead. There is some room for growth, especially after the watershed of 1870-1890 when the Long Islands in particular can be expected to surge in growth and population. But hard core British subject loyalists will have to either make do with staying home or find other niches such as Cape Colony in South Africa, Australia, and New Zealand; perhaps the RN (without describing how, I take it as given that the British will not get a foothold in the Pacific northwest) will seize Hawaii more aggressively and earlier. (USA still seems likely to get Alaska though).
Against this, some Britons emigrated precisely to escape the monarchy and British institutions generally. These might have been fully absorbed by the USA OTL, but perhaps more would take the step if the USA were larger and more secure.
Meanwhile we have a lot of people, from Britain and from other European nations as well as especially later, other British possessions in the colonized Third World, who left their homes essentially for economic reasons, willing to go wherever they could hope for land or anyway fresher opportunities. The lion's share of these did come to the USA; others to other settler colonies. All of these who went to Canada OTL would probably go to the USA here.
Thus overall, I think it is likely that in terms of European emigrants, the USA absorbs a number comparable to the US and Canada OTL combined. Lack of British loyalists is made up for by encouraging more anti-loyalists.
However, I certainly have to agree with another poster above on the question of "nonwhite" immigration--the USA will not be welcoming to immigrants from British possessions who are not European, "white" to American eyes. We might hope for some moderation of American racism in knock ons I am tempted to outline, but these are a bit farfetched honestly and probably American exclusivism prevails. Now there actually weren't a lot of effective restrictions prior to the early 20th century, those we had were specifically targeted--prohibitions on East Asian immigrants for instance which evolved in the American West. I am aware of SCOTUS cases involving making judgements on the racial status of people like Turks and Armenians in the era before the immigration restrictions were sweepingly imposed. This shows at least some such people immigrated without anyone stopping them. So in fact a certain number of South Asians for instance might find their way to the USA who OTL headed for Canada. Less than OTL I suppose must be correct and the door slams shut in the early 20th century, hard.
The question remains--do the territories of OTL Canada get more settlers and growth, or less?
Overall I think the trend leans toward "less." More of the net equivalent or nearly so collective immigration will settle south of the OTL Canadian border. The exceptions relate first of all to Quebec's status as magnet of specifically Francophone immigration, and possibly dominating settlement of OTL Ontario.
I suspect Scandinavian immigrants might be shifted a bit farther north, straying into OTL Manitoba, Saskatchewan and Alberta (this latter having a different name in the ATL obviously) but by and large, the farther north the land, the later it would be settled. It is possible that tracts of the more northerly fringes of cultivatable land would be reserved for Native Americans and maybe Uncle Sam actually keeps the deal for once--which might mean more Native American people surviving, but this seems unlikely to offset fewer "white" settlers. The lands north of OTL North Dakota and Montana I think would be settled later than these plains states would be--to be sure, the concentration of some of the OTL emigrants from Britain and elsewhere who went to Canada might accelerate the settlement of the Midwest and Plains versus OTL clocks, so that say North Dakota is admitted as a state a decade earlier, and thus Saskatchewan not long after that. And such expansion by agricultural settlement will slow down and thus leave the OTL provinces less settled overall.
As someone else points out, it is possible some major Federal government initiative will seek to promote settlement of the northlands, akin perhaps to the TVA project of the New Deal.
This brings me back to what I think the likely borders to be--I deemed it unlikely the USA would claim or get Rupert's Land, Labrador or Newfoundland, either in the ARW settlement or in some later clash. By the time the USA develops the reach to muscle into these territories, the will will have ebbed away. Britain thus will in outcome continue to claim these far northern and northeast coast lands, and might or might not claim the Yukon as well.
If this is the case, then during the uneasy sometimes hostile general peace of the first century, it will behoove Federal authorities to at least try to set a watch on Rupert Land's sprawling boundaries, so a string of frontier forts seems to be in order. Such forts would not be easily sustained, but efforts to try to encourage some settlers to provide a market source for rations might be made quite early, and insofar as goods must be shipped in, or funds to purchase local supplies appropriated, this helps catalyze private enterprise growth. In turn, this might make local ventures at mining and so on more viable than OTL. Now after the turn of the 20th century, such border guarding no matter how skimpy would seem more and more frivolous and questionable. But at any rate a basis for infrastructure, roads and railroads and telegraph lines, would have been laid down; the border towns might diminish and go to ghost towns, but in some zones anyway one would think sustained enterprise might prevail, particularly if there is a series of mining operations. Also as relations with Britain improve, north Canada Native people might strike up trading relations with the Yankees across the border, which could become the lifeblood, or anyway a critical catalytic safeguard, of the viability of other communities.
This depends on ignoring part of the OP premise of course and supposing the tier of northern OTL Canadian territories remain British, along with the far northeast Maritimes and the weird annexation of Long Island.
Overall I expect that despite such special countertrends, the center of combined population is well south of OTL, the tier of land north of the USA border OTL has fewer people despite perhaps a legacy strip along the Rupert's Land/Labrador frontier.
Perhaps Quebecois and maybe Louisianan birth rates would be higher and thus result in a net overall gain, enough to make Quebec itself, northern Arcadia, the land known as Ontario and an infusion of generic higher settlement of the Plains great enough to shift that center north again and populate the overall region of Canada more highly than OTL-though I would think then the detailed pattern would be different--heavier in the east, lighter in the Plains.
I certainly think if the sites of Seattle and Vancouver City were under the same administration, the latter would be favored by economic and logistic considerations and eclipse the former, so basically shifting part of OTL Seattle and other northern Sound regions to aggrandize Vancouver (which might or might not have that name) would amount to a local northward shift, as would US colonization of Vancouver Island. But overall this is not a large effect versus the entire combined US/Canada population.
Long Islands I expect would be pretty densely settled and if British Commons apportionment is at all fair, the great urban complex at its south would have dozens of borough seats in Westminster, while the country ridings (many becoming boroughs too) of the lsland would also be numerous, including an ATL borough area in the far north, a legacy of the days when the Empire felt it needed to defend the Island foothold in depth.
Nova Scotia however I expect would saturate at OTL levels pretty much, along with Newfoundland and Labrador and the northern "Rupert's Land" tier.
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as a general thing, without the clean break of most of Canada, or anyway the largely inhabited part, falling to the USA long before anyone European had settled much of it, I'd rather not face the acrimony involved in setting up a scenario for Yankee conquest of largely loyalist Canadian populations at some future date. The peaceful alternative is imagining some kind of voluntary plebiscite in which the Canadian Federation votes to apply to join the USA and the USA agrees to thus aggrandize itself with this windfall. This is however close to ASB, politically speaking. Why would Canadians do that? Only this or an ATL conquest of Canada after 1900 puts the POD in the right time frame, and the politics of conquest greatly overshadows the rationality of settlement patterns.
Fundamentally I'm endorsing the idea that the full extent of modern OTL Canadian population residing there and not on current US soil is an artifact of that political division, confining immigrants who chose to go with Canada to choices north of the border, where if that border did not exist I guess more of them would have chosen to settle farther south. Vancouver versus Seattle being the big but unusual exception.
