Given infinite wisdom and/or the benefit of hindsight, one can wank any nation one chooses into total world empire. Given instead realism in these matters, the inevitability that mistakes will be made, that short-sighted immediate interests will cut off long-term possibilities and that enemies will be made, imperial overreach and failure is inevitable.
Britain's best chance, had they had this sort of infallible foresight and kept to a parochial idea that they must above all maximize British control, was of course the original American Revolution. I'm not sure if it would have been possible for Britain to expend more lives and treasure than they did, realistically, to try to bring the rebels back in line, but they certainly could have on paper. The question would be, would keeping the British government on track on a firm anti-rebel policy while at the same time adequately safeguarding their interests in Europe and in other colonies have overstrained the attempt and led to some sort of breakdown? I infer, from the fact that London did come to terms with the rebels in America, that yes, Britain's leadership collectively did fear such a risk. But perhaps if they knew of the humiliations the once-revered Empire would face at Yankee hands in the middle of the 20th century, might they have stayed the course and won?
Say they did. What would be the outcome? On one hand, the imponderable effect of the American patriots achieving their goal on republican terms would not exist. How much bearing did that have on the secession of most of the Spanish Empire from Madrid's control did that have? On the rising of the French people in 1789? We don't really know; arguments can go either way. Leaving the state of the Spanish world and the possibility of the rise of Napoleon aside for the moment as a triumphant Tory regime in North America surely would have, what would the Loyalists and their leaders in London do in BNA?
Probably enforce pretty much all the measures that provoked the rebellion in the first place, and notable among them, the prohibition on British settlement of the northern Mississipian lands west of the Appalachians. For two strong reasons--on one hand, colonial authorities had been realizing for some time as 1776 approached that the colonists derived a lot of their ungovernable attitude from the feeling that they could always move west and try their luck again; the prohibition of westward migration was in part intended to bring them back into line as properly loyal and dependent British subjects. Also, the King had made treaties with those Native American peoples who made peace with him in the wake of the French and Indian War; he owed them protection, and they would hardly be safe if an unstoppable tide of ambitious settlers headed their way.
Thus, the expansive potential of the United States of OTL would not be fully available to a British-ruled set of colonies. Not as early anyway; British colonial policy in Australia and Africa suggests that sooner or later they'd succumb to the temptation the vast wealth of those lands, and lands farther west, seduced them with. But they'd still have to worry about what a huge population of settlers, even ones whose ancestors came from Britain and who had been carefully kept more or less loyal hitherto, would do once they were so far from established centers of control and began to outnumber the population of Britain itself.
A Britwank that preempts the USA completely then, would be one in which the net economic power of the USA of OTL either does not exist at all, or exists in fragmented form, divided among many powers. Thus while North America is an asset and in no way a liability, it is less than OTL--Britain can't claim the whole thing for her ledger. If perchance settlement and growth of the North American continent under British control does go forward anyway, it seems inevitable the political center of gravity must also be drawn across the Atlantic with it, or else some kind of belated rupture severs them.
Or of course, the British show uncanny guile and craft, and somehow keep the massive new Greater Britain subservient to the dank little island it was colonized from. How likely is that, and how can it be done at all?
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OK, so suppose the British must as per OTL grudgingly let the USA go, but do not adopt any policy of conciliation. Well, for the most part, that looks rather like OTL right up until the War of 1812 anyway! Does it not? As it happens my favorite Britwank is Dathi Thorfinnson's Canadawank, a project I discovered after it died, badgered the poor man into a brief revival, then when serious RL issues led to a hiatus, died again--I have to sadly conclude for good. A somewhat better showing for Canada in that war, leading to a second war deliberately launched by the Yankees decades later, leaves the USA a pathetic shadow of its OTL self (but still, as Dathi pointed out, quite a substantial power, if a humiliated one) and British North America, as a loose patchwork of diverse realms, mighty and yet properly deferent to British authority. As of say 1850 anyway--would Dathi's mighty greater Canada, with holdings in California, a greater Texas and another Mexican province as well, and indeed everything west of the Mississippi, remain a mere vast province of an empire centered on London for centuries to come? There is some reason in the timeline to think that maybe it would.
But enjoy it as I do, I can't say it seems probable that the British would always step right and the Americans always step wrong. What seems more likely, in the event that evolving events lead to sustained hostility and suspicion between the Yankees and British subjects, is that the Americans would do rather better in holding and expanding their territory, and develop, with far greater determination than shown OTL, a military establishment, including a Navy, that the British would have to reckon with seriously, to their detriment elsewhere in the world--perhaps Britain would be distracted from consolidating her hold on India, or fail to dominate Egypt and thus communications with India. Perhaps the Russians would menace British interests with greater effect, or a Franco-German alliance would form, while Britain is distracted trying to protect Canada and her Western hemisphere interests in general from Yankee agression.
By 1850, if the USA could accomplish more or less the gains we'd managed to by then OTL, I'd say it would be beyond British power to break the United States. They could cut us off from much world trade but we weren't strongly dependent on it. They would as a matter of course cut off most of the foreign investment that enabled US development--but under that national emergency, I believe the US government would respond by fostering internal reinvestment, perhaps at some stringency, in the name of the manifest national interest, and more or less match OTL levels of growth--whereas Britain would not receive the considerable return on her US investments she enjoyed OTL. An unchecked Royal Navy could wreck havoc on the American coasts and launch invasions inland--but as circumstances darkened in that direction, Americans would build up our own navy, and fortify the shores, and counterattack, to a degree that might put British control of the seas in doubt and thus encourage nations like France, German states, perhaps even the Dutch or the Russians, to pile on and enjoy spoils of their own.
Such an aggressive, militarized USA as this might not enjoy the same immigration levels and population growth, nor the same degree of per capita prosperity, that made her the mighty power of the 20th century we know of OTL. But on the other hand, being engaged against the strongest single power of the age it would surely be engaged in alliance politics, drawing still-considerable force in on the side of European conflicts and quite likely, as long as Britain maintains a hostile stance, severely whittling down British power.
Now compare that to the "conciliatory" policies of OTL. These bought security for Canada and other Western Hemisphere holdings at a very cheap price. They gave Britain access, as investment partners, to the vast expansion of US wealth. They tied American interests loosely but strongly to British interests; Americans refrained from challenging her at sea (until a time came when the RN was rather grateful for the extra strength American alliance could give) and made no aggressive moves for a share of global maritime trade as they had in the early, competitive years, leaving sea trade largely a British monopoly with only token competition. Until the 20th century American misadventures in Latin America south of Mexico were few and desultory compared to the systematic British informal domination of most of the hemisphere; the Monroe Doctrine served largely as a cloak for this British free hand. Americans made no attempt at grabbing any part of Africa (except insofar as Liberia could be seen in that light; a pathetic and second-hand attempt indeed) and only token bids in Asia and the Pacific. Dominance there was largely a game for Britain to play against second-rate French and third-rate other European efforts.
Then we come into the twentieth century. It is quite true that American ambition expanded in this time, and efforts were made at British expense--but it is also true that Britain got into serious trouble without any Yankee schemes being to blame, troubles inherent in imperial overreach without the benefit of superhuman, ASB foresight and cleverness.
Suppose that instead of looking at 20th century Anglo-American relations through the rosy crystal of the "special relationship," we take a jaundiced view instead, of ruthless Yankee encroachments on the glorious Empire. Suppose indeed that that view had prevailed in London, the moment the Americans seemed ready to step forward?
How could Britain have fared had she not been willing to mortgage her holdings to Yankee creditors? Had she been too proud to ask for American help in winning the Great War? It seems certain to me that the Americans would have held aloof, watching the Germans prevail over France and leaving Britain, battered and bled, to make what she could of her overseas power in denying them colonies while the Germans remade the Continent to their own convenience.
Drained, demoralized, but with a free hand in the colonies, having gobbled up German ones and probably French ones too (to the disgust of their Entente former ally) and owing Washington nothing, could Britain have kept her balance and recovered from colonial holdings alone, while frozen out of continental markets both in Europe and North America?
Britain's colonies, at their height, broke down to two categories--the "white" settler dominions of Australia, New Zealand, and Canada, and colonies where a small British-blooded elite attempted to dominate native peoples who vastly outnumbered them--this was true even in the intermediate case of South Africa, where whites (divided among loyal British immigrants and prior and disgruntled Boers) were outnumbered ten to one. The ratios were even more meager in the rest of Africa and India.
Assume for the moment that there is no centrifugal tendency of the distant settler colonies to go their own way, that they will all muster at the call of London and the King without question, that they will not count their numbers and ask why they are the tail and Britain is the dog. Indeed by numbers the British dominance has a point--add up Australian, New Zealand, and Canadian populations and you don't have a third of the population of the USA--rather a bigger proportion of the British population but not quite, or barely, a majority. The point here is, counting all of these as solidly "British" as one, and we still can only just match American population.
So on paper Britain clearly comes out ahead, because in addition to these appended nations, she also appears to control an order of magnitude more people, her non-European colonial subjects. But just what worth are they to British power?
They comprise a vast market, and cover lands that contain vast resources, and have in fact, OTL and in the history of this ATL, yielded up soldiers who have fought for the Empire.
But the terms on which they are its subjects are demeaning. To tap their potential, the British must simultaneously exploit and conciliate them. It's the same problem our hypothetical victorious King George III faced with the vast prize of Western North America--a prize he dare not plunder lest it turn his native allies against him.
Well, in the Empire as we knew it OTL, the die of white supremacy had been cast. Could the Empire turn over a new leaf at this point, and start incorporating its subject peoples on more equal and dignified terms, to get them accept the identity of British subject with grace and enthusiasm, and thus make good the vast potential of their global empire? Doing so would have raised discontents among the white British, especially the very ones who went south to the colonies to hold and exploit the place for their own betterment (and the good of the Empire of course!)
If instead the empire took a hard line of white supremacy, I suppose that as the Apartheid Nationalist government of South Africa did OTL they could prevail on those terms for some time, but at the cost of raising up vast discontent against them.
If at that juncture the USA were indeed the predatory thug the OP seems to imply by referring to British policy as "conciliation," the Yankees, and other rival powers such as Germany, France or Russia, would have opportunities a-plenty to assist in disrupting British control. Doing so would make them all big hypocrites of course but the subjugated peoples might well rationally prefer to see two or three rival hypocrites wooing them for favors over one smug one that feels invulnerable.
At the end of the day, I don't think it's reasonable to blame the British failure to keep control over the vast majority of the world on the schemes of her rivals. The problem is inherently hard to solve and the only fair solution would have had the British Empire evolve into an Indio-African one, with the white English speaking lands being a mere minority faction in something vaster and distinctly non-European. That might be a fine thing but how likely is it? Far more likely would be a hostile USA that doesn't owe Britain anything joining in a general feeding frenzy.
OTL I know there are British perspectives that see the USA as having accomplished just that, on its own without sharing with anyone. It still seems an odd perspective to me, even acknowledging the degree to which Americans did indeed plot for supreme power, to a large degree at British expense. And anyway, it seems clear enough now that the USA's day as a hyperpower able to get whatever it wants whereever it wants have already passed--we can surely throw tantrums and mess things up anywhere, but getting what we want is not so easy.
I therefore chose the option of "Britain would be worse off," figuring that while the British did not have perfect foresight, they generally made pretty good decisions considering the situation they actually faced, and those decisions gave the collapsing Empire a soft ride down and left Britain herself, and the Commonwealth nations, quite well off and distinctly favored on the whole by American power when it was dominant, and in a position to align with American policy where it suits them and abstain from it where we are misguided today.
The second-best is that "both US and Britain would be worse off," but I'm enough of an American patriot to think that had Britain been pig-headed enough to throw down a challenging gauntlet in the mid-19th or early 20th century, despite the liabilities of taking down a big and tough adversary, we might have come out at least as well off as OTL. I prefer a world of Anglo-American friendship myself, and trust that it is probably the best outcome between us.