But with a POD after 1945, can Britain possibly be viable enough as a Great Power for the "split" to have serious consequences?
Having posed the question I can see it is badly posed; if the US/UK relationship has gone sour, presumably the US/French relationship is even more bitter, and NATO is disintegrating completely. The USA still has legalistic claims on bases in West Germany, but how will the Americans access Germany if both Britain and France are freezing us out? Anyway with considerable uncertainty as to how Britain and France will react if things come to a head on the East/West frontier, Americans are more likely to throw in the towel and say, "Europe, you're on your own."
So--even granting that the UK loses its colonial sweep, all the faster due to lack of American de facto support of the fragments of Empire, the "split" has drastic consequences for American roles in Europe.
Unless in reaction to the split of the Anglosphere, the French swing over to a more pro-American position--not so likely if the POD is based in Suez since the French government was as badly screwed in their perception anyway as the British by US policy. But if the POD is something else entirely--the "split" then means that Britain unilaterally abdicates as a world power and spirals off into impovershed irrelevance. This seems unlikely to me; someone in Britain would speak up for maintaining alliance, someone in America would do the same. The POD really has to be one nation or the other putting real monsters of arrogance and stupidity into power--well, I can't unfortunately rule out that possibility in the USA!

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World of Laughter, A World of Tears, for instance--and tragically, Walt Disney isn't really a monster, even Roy Cohn isn't--but their style of conservatism seems all too likely to unleash real monsters.
It seems more likely for the stupidity to come from the American side than the British, since the British Empire, such as it was, was on life support from the American side and the Big Englanders knew it in those days, while a Little Englander had to be more in favor of more cooperation with Europe, which would also proceed under a US aegis, holding off the Soviet threat.
The odd pattern I've noticed in post-war European relations with the USA has been that while in Germany, as one might expect, the ascendency of the fairly conservative Christian Democrats has coincided with a more strongly pro-US policy while the rise of SD and other leftists has tended to make for a more arms-length relationship with the US, in France and Britain it was generally rather the opposite--those periods when France was most defiant of American hegemony, and Britain most committed to an independent policy, were when they had Conservative governments, whereas the rise of Labour/Socialists in the respective countries were the times of most close collaboration between them and the Americans.
So--a formula for a US/UK split seems to boil down to either the USA going completely radical--either to the right so very far right that even British Tories can't stand it, or going leftist at a time when the Tories are strong. Well, if one can imagine the USA going very far left in the immediate post-war period, which is hard for me to do--the elements are there to an extent, but the elements that OTL crushed them seem impossible to butterfly away without a huge civil war, one where the progressive side would start out dangerously weak--that might conceivably lead to a Tory regime determined to hang on to the Empire--but how could such a regime defy both the Labour sentiment in the UK itself that would almost certainly want to hitch it's wagon to the leftist Yankee star (in partnership with everyone's good friend Comrade Stalin of course!) and the combined military potential of said Stalin, the Yankees, and the determination of colonial peoples to free themselves, especially with all these powerful leftist regimes offering to help them free themselves?
The reality of the post-war world was that the USA was riding very high in the West; both French and British conservatives had to realize that they were weak enough both domestically and internationally with the half-baked, left-handed, inconsistent support their colonial regimes did have from the USA and they didn't have a leg to stand on trying to defy the Americans head-on (at least not until the 1960s).
If the alignment of leftist versus rightist domestic politics were somehow different in Britain and France--if the leftists there, Labour and Socialists, had early on felt that the Americans were a big problem but on their own they could reorganize domestic economies and also rally support for their respective empires among the colonized peoples with credible promises of a new order in mutual cooperation for everyone's mutual benefit, while the USA went farther right--that might conceivably do it, except that the Europeans were terribly weak and vulnerable in the 1940s, and their colonialist elements were very unconvincing to the colonial peoples.
So maybe, if French colonial policy could recover credibility among peoples like the Vietnamese and Algerians very soon during the early '50s (just about impossible but maybe someone could lay out a roadmap?) and the British Labour governments could be more effective than OTL at developing economic recovery and advancement on a socialist basis, one that benefited colonial peoples who also enjoyed advancing civil rights as well, thus holding both Empires together on an increasingly populist basis--all this with the cooperation of the Americans up to say the mid-1950s--and then while the Euro-socialists and their constellation of imperial possessions are riding high, the Americans go pretty far right--doubling down on American segregationism for instance, going even farther than OTL with McCarthyism and repression in general--I could see a general US/Europe split around 1955-60.
I've been lumping in France because even if Britain alone could hang on to her colonies, most of them anyway, and develop a strong economy based on high tech and integrating the resources of the Empire into growth and progress for all, Britain alone could not substitute for the USA in NATO. They'd need to pool with France as well, and be drawn into a strongly European-centered policy as opposed to the usual British overseas orientation. With Britain and France together, on a leftist policy basis that however firmly refuses to be drawn into the Stalinist system, West Germany and Italy would remain attached, the various smaller European nations would also remain in the constellation and Western Europe as a whole could be standing independent of a right-wing, reactionary USA.
Maybe someone else can see a way clear to some other configuration that could be stable; I think if the USA goes socialist it would take a post-OTL-1945 Western Europe with it; a Tory Europe that is not de facto aligned with the USA however defiant it is rhetorically is doomed to collapse IMHO (De Gaulle kept France in NATO for instance).
A strong and progressive Europe could well defy American policy vis a vis Vietnam--but of course the way I see France surviving as an equal partner to Britain, Indochina would still be affiliated to France anyway.
That's the part I think is rather ASB--France had practically zero legitimacy in Vietnam and Ho Chi Minh, who might have reached some accommodation with Americans but certainly would not with France, had quite a lot of it--maybe I should rethink France's role, have the French try and fail to retain their empire while Britain keeps hers and instead of De Gaulle, some other Fifth Republic joins a British-led European socialist-progressive federation--perhaps the British coax the former French zone back into an alliance relation not with France specifically but Europe as a whole and the French do play a special role in developing multilateral industrial and social relations with those countries.
In that latter case, Vietnam could still be the same sort of battleground it was OTL, with the French, before their conversion to socialistic globalism, losing the North as OTL, deeding the South over to the Americans rather than to Europeans, and 'Nam becomes a cautionary tale about the vices of the conservative/American approach versus the progressive European one.
Maybe the French learn enough wisdom soon enough not to lose Algeria as well, or lose it to a movement willing to become an equal partner in the European progressive federation?
I'd think we'd need multiple PODs, many of them rather unlikely, to make something like this happen. America going far right is not that unlikely unfortunately, but I don't see it automatically forcing the Europeans to become more successful--rather the opposite! Someone who knows a lot more than I do about the nature and details of European, particularly British, politics would have to outline how Labour could have been more successful in reviving British industry while also shifting the benefits of progress down the social ladder to solidify the stake of working people both in Britain and overseas in the colonies in a socialist system--many here of course think the whole problem Britain had in the post war period was too much socialism! I think they are wrong and with any less of it Britain would have collapsed completely, and more of it would be a good thing if intelligently done. The whole argument hinges on whether "intelligent socialism" is an oxymoron or not of course! OTL in my understanding much of the credibility the post-war Tories had up until Thatcher anyway rested on co-opting Labourite remedies in a way that didn't scare the big moneyed classes too much, but clearly by the late 1970s that approach had reached an impasse which Thatcher broke through by radical means that appear to me to have hurt a lot of people.
So the anti-socialist crew can feel free to make their pitch for a tough, competitive Britain. Good luck.
I think one way or another, for Britain to have the standing of being on one side of a Western "split" with America on the other, the UK needs access to resources and markets on a global scale, and I've outlined the only strategy that occurs to me for retaining at least some of the old pre-war preferential access they had during the Empire days. I don't see how earlier Thatcherism could accomplish that, not in the face of American competition. American competitiveness could fail if the USA goes into a dark enough phase; conceivably if the USA is aggressively white-supremacist the British might be able to persuade her colonies and those splitting off from say France that they need the protection of a strong power that isn't so insane on the subject--but that's one reason I think they have to be socialistic; the old British colonial regime might not look all that different from US white supremacy to Africans and Asians.
Is this split, even granting my moving European politics and economics leftward, plausible if the USA doesn't go nuts? Less so certainly; part of the credibility of a progressive regime is that it tries to reach a decent solution with rival powers by peaceful and friendly means, if the Americans are being reasonable then we wouldn't get a second, European-led bloc, we'd get a smoothly running collaborative Western order of the kind I was raised to believe we'd achieved OTL!
Such a split-up implies a rather miserable failure on the American side then.