I think Native peoples would continue to be "shafted," but not quite exactly like OTL. As RougeTraderEnthusiast says, it will depend on circumstances; some peoples will be wiped out as ruthlessly as the USA ever did (which rarely results in the total death of all people in a tribe unless it is a tiny band, but does end in cultural genocide as the few remnants of a group intermarry and thus are stricken from the count of the group's survival, and their distinctive culture is pretty much forgotten) but others will get a considerably better deal.
I believe that the prohibition placed on colonists moving west over the Appalachians was one of the major drivers of the rise in popularity of revolutionary sentiment. And that some leading British authorities had the impression, which I would say is correct, that the perception that anyone could always abandon their position in the eastern settled parts and go west to the frontier to try again was a major cause of what looked to them like indiscipline and disrespect for class hierarchy, indeed for any accountability to the larger British system at all. Restricting British American colonists from expanding farther west was in fact recommended policy of some officials put in charge of trying to retain American territory; the hope was that then they'd come around to a more properly British respect of appropriate stations.
This is in fact the largest single reason why I believe conflict was inevitable and would happen very soon, not being postponable by many years let alone avoidable. It was against leading British interests to allow either of the two major paths by which British American colonists could get rich--expansion onto new territory, or entrepreneurial competition on an equal basis with British-based enterprises. The third path--prosperity by agricultural enterprise that is complimentary to British production, producing crops that cannot be grown in Britain, was supposed to be the first and only way according to colonial theory, and could make someone quite rich--but with the expansion of settler territory checked, this benefits only the few who happen to already own the most suitable land. Thus the drift of British policy for the north American colonies was to impose an aristocracy characterized by dependency on Britain, and to stunt the economic growth prospects of all others. This is what Americans were so angry about. And the British perception that simply letting the colonists do what they wanted to would surely lead to more insolence, more democratic loss of deference, and money-losing conflicts of interest was also correct.
I believe George III also very sincerely felt he had a moral and indeed legal obligation to protect the interests of those Native peoples who had submitted to his rule. Their and his negotiators had made a solemn treaty with each other and Britain had to uphold its side of it in order to retain basic credibility. It was plain to anyone by then that unchecked interactions between settlers and Native people always led to great grief for the latter. A blanket prohibition on settlers entering the region where substantial Native tribes still survived was necessary to protect them even if it had not also been recommended to hem in colonial expansionism anyway. To be sure also, particular British interests, such as fur traders wishing to operate along the same sort of lines as the French had, or the Hudson Bay company did, recruiting Native people to bring in the goods to trading posts, also existed or wished to exist, and had the ear of British governing circles too.
So, with the conflict inevitable, the only way Britain preserves sovereign control of BNA is to win the Revolutionary War, or rather in the language of the ATL, quash the insurrection and keep order. Since this will be a major hurdle, the investment of military repression will be large and will leave much resentment behind, therefore the major point of contention, the freedom of colonists to expand west (and engage in productive enterprises such as factories or world-wide merchant shipping as nominal Britons) versus the imperial interest of Britain to keep them in a subservient, useful but not threatening status, would be sharply outlined, and the imperial side will have won. Furthermore, the Indians (to use contemporary terms) will have sided with the Crown against the colonists. The Crown now owes them an even greater obligation, and lest the King's men be tempted to forget it, needs them still to help hold resentful colonials in check. It was after all British policy after conceding defeat and recognizing the USA in OTL to continue to try to check American ambitions by supporting the Indians in the territory between the Mississippi and Appalachians they had legally conceded to the USA in insurrection against Americans OTL, a major issue leading to and in the prosecution of the War of 1812. ITTL with Rebel pretensions to self-government crushed and Britain's and the Crown's rule prevailing all across the territory to the Atlantic, surely the Crown will act to protect Indian interests, and prioritize only those enterprises that do not conflict with Indian interests, such as a fur-trading monopoly. The Indians for their part will quickly come to recognize the Crown as their advocate versus rapacious colonials, and sound the alarm should any disgruntled would-be revolutionaries decide to defy the law and pull off wildcat settlements comparable to the South African Voortrekkers.
Now in the long run, the Indian position in the west would continue to deteriorate despite major advantages over OTL. Eurasian diseases will continue to devastate them; with the best will in the world 18th and 19th century European medical "science," such as it was, would hardly be able to do much to help them. The main protection would be isolation, but a certain degree of contact is necessary to preserve their status as Crown subjects and supporters. Therefore peoples once numerous enough to effectively occupy vast territories (especially with the support of British armies) would melt away to remnants, and large tracts of land would appear to be completely vacant. Versus the colonists the Crown needs the Indians, but they are declining, whereas the basic greed that led to the original foundation of the colonies in the first place remains in place, all the hungrier as industrial development proceeds in Britain. Sooner or later, the Crown will permit carefully vetted colonists to move into the West, and negotiate the withdrawal of even strong and closely allied and somewhat assimilated Indian peoples into more compact territories sized for their diminished population. The west will become a patchwork of British settlements (largely drawn from colonial Americans, despite the desire to Anglicize them for maximum loyalty and control--after all, about as many Americans were Loyalist/Tory as Rebel/Patriot OTL, with another third of the population trying to sit it out with as little commitment to one side or the other as possible, so despite added resentments in the wake of harsh British measures after winning the war, a lot of American colonials will be of proved loyalty to the Crown too), largely assimilated Native territories that probably will have a lot of European settlers among them, and more xenophobic tribal territories attempting to live more or less traditionally. These latter will be in a weak position on several grounds and tend to give way to settlers or the more engaged tribes.
I think there might be an interesting phenomenon for the British system as a whole, that the Crown itself, as opposed to the Westminster Parliament, might have opportunities to secure revenues and even manpower recruitment bypassing Parliamentary control. Someone more familiar with the legalities of the Westminster parliamentary system and the Glorious Revolution might be better able to rule on this--if Parliament automatically gets control of every revenue the Crown might claim, this won't happen on paper anyway. But if Parliament was merely given control over existing taxes and excises, perhaps in the course of the emergency putting down the rebels and then keeping order in the colonies, the King's people get control of revenues (a portion of the fur trade profits for instance; tributes paid by Native peoples for protection, punitive fines on the Americans for rebelling) that never get routed to the Exchequer but instead stay in the hands of royal appointees. Parliament might legislate terms on which these agents ought to act, but as the King's own agencies perhaps their de facto independence, underscored by Royal majesty, would mean it is more a negotiation than a command relationship.
In the short run the King would definitely need to somewhat centralize regional command in some viceregal capital, but I think that the effect of having to quash American rebellion would tend to recommend that as much power as is practical gets centralized in Britain. An elaborate colonial administration might exist, but legally subordinated to policies set in London. As communications improve over the 19th century, with railroads, telegraphy, and steamships, ultimately a transAtlantic cable, centralizing control in Britain becomes more practical despite the vast rise in American populations to be controlled. In terms of legally recognized agencies of democracy, policy will be divide and rule--create lots of small regional bailiwicks such as the pre-Revolutionary colonies, but all local legislatures are subject to override by the British Parliament and/or Royal decree. Native tribal governments will fit in this same framework--they would be legally protectorates, their self-government under Royal, which might or might not mean British Parliamentary, direction. No democratic overarching body coordinating the many colonial or protectorate regimes would be allowed; any element of democracy challenging Royal administration would have to take the form of petitioning Parliament to act on their behalf. Would the British Parliament come to include representatives of the various colonies (not just North American, but also South Africa, Australia, NZ) elected directly? If British subjects resident in Britain are not to see their many MPs reduced from a thousand to mere hundreds, it would be necessary for Parliament to be greatly expanded, to many thousands; I doubt this would be practical. Perhaps overseas (white!) subjects would have to content themselves with a very fractional representation, with a handful of MPs elected by far larger electorates than in Britain, and Parliament expands only modestly, to 1200 or 1500 or so. (That would neatly solve the problem of how to include India without being overwhelmed by it of course. At the cost of the vast majority of British subjects, white or otherwise, being terribly underrepresented). Racism could also come into play. Another big topic for this thread is how slavery gets handled on an Empire scale including the American south, but here I might suggest that anyway neither people deemed "black" in America nor Indians, and thus perhaps not Metis, are considered for Parliamentary representation at all--perhaps instead parallel agencies such as a Council of subject Kings has standing to be heard in Parliament and via their treaty relationships with the Monarch a practical alternative input into Imperial government. Assuming slaves are ever freed, they might get folded into "white" America for purposes of voting, might simply be denied any franchise at all, or might get some parallel representation, perhaps through an administrative Freedman's Ministry or some such.
I expect whatever emerges to be a bewildering patchwork of historically based regions with special privileges and obligations; the principle of uniformity in rights and power would not strongly apply, with everything being solved ad hoc.
The question of what happens to Native peoples also relates to the question of what geographic expansion, if any, would BNA have. Is Britain sure to seek to grab territory from Spain eventually? Perhaps not; the contingencies of British diplomacy in the 19th century OTL put them on the side of Spain more often than not, at any rate on the Spanish monarchy in exile's side. It was generally British policy, from some time before the ARW and long after it, to pursue her interests in Latin America by means of getting acquiescence for British trade in Spanish ports, including handling the slave trade from Africa to Spanish colonies, and later to get compliant Latin American governments that would again mean informal British hegemony over nominally independent republics.
Now this world is majorly butterflied by the defeat of the American rebels in BNA of course! Many would assume there would be no French Revolution at all in such a TL. I'm not so sure though; the notion that the French Revolution was triggered by the American one seems questionable. Certainly the major debts the French monarchy took on to assist the American rebels (and pursue French interests at British expense all around the world) were a direct factor in the chain of events leading to the OTL 1789 uprisings and eventually the First Republic. But it is entirely possible that the French will go all in if the Americans rebel, even if they are eventually defeated. The lack of an example of a successful republic overseas might be offset, or more than offset, by the French being on the wrong side of the war and the disgrace that might bring the monarchy.
Without a French Revolution, the factors that brought Britain and Spain to the same side would not be in play; the Bourbon Spanish monarchy would be seen as a catspaw of the French one. Britain might then perhaps favor Latin American uprisings and declarations of independence, but that would be a dangerous game to play considering the sentiments it might stir up in North America. Britain might instead offer to support a Spanish empire plagued by uprisings in return for territorial considerations.
We'd really have to game out European events carefully. After all with no French Revolution, would there have been sufficient drive, over the long run, for a fully and fairly democratically representative Parliament for Britain? If not, the whole question of equal and fair representation would be mooted; a handful of American MPs might seem like progress to Americans rather than humiliation; the House of Lords might retain considerably more power, and be a more suitable instrument for pragmatic representation of strong (that is, rich) colonial interests, even those of Indian (south Asian subcontinent and American Native also) potentates and millionaires. Empire policy might slip over to the Lords even in Britain.
The very idea of universal democracy might seem as strange, outlandish, radical, and Utopian as the abolition of private property, all across the European ruled world.