Lots and lots and lots of colonial uprisings, bush wars and out-and-out national liberation.

I don't think the UK has the capacity for the kind of long-term push and pull that the USSR and USA had OTL. Whatever flair ups there are, the UK is almost definitely going to be on the defensive even if the appearance of conflict is a stalemate. There's just more vulnerability to exploit for the USA than vice-versa.

So long as the "Cold War" is just a by-word for semi-hostile diplomatic maneuvering, then I think the UK and USA can remain at parity. But the moment things actually start heating up, the millions and millions of restive populations that the British are sitting on top of are going to blow up in their face.

Honestly, I don't know how plausible it is to have the UK and USA both to have leadership that are committed to this kind of antagonism over a long period of time. The British have too much to lose and the USA doesn't have much incentive abroad beyond its commercial interests. Collaboration between the two powers has historically been very profitable for them both.

If you want this kind of longstanding conflict to have legs, you need to have a US with very different internal politics. One that sees opposition to Britain as a cultural virtue with a value that's beyond what's practical and impractical. And you need to have a UK that is responsive to that hostility in kind.
I don't think is possible with a POD after 1900, just because there's not enough cultural steam behind any kind of Cold War without it. It can definitely happen, but there's going to be a great deal of internal opposition in both countries to the state of conflict.
This is sort of a sequel to a thread I posted earlier.

Long story short is, what would be the effects of a Cold War that pitted the US vs the British Empire.

It all depends on your PoD.

The UK on its own could have a larger economy than the Soviet Union (it didn't always in OTL, but if it is run well the potential is there), the UK plus its settler colonies at all points of the OTL Cold War had larger economies than the Soviet Union, so a UK that managed to form some sort of imperial federation in the first third of the 20th Century could easily be a peer rival to the USA. Depending on the PoD, one could have such a British federation be a little stronger than the US or a little weaker with fairly small changes to either.

A Britain that retains all or most of its empire (which requires either no WW1 or that Britain becomes much less racist very fast) would be a beast that were rightly feared.

Additionally, one could easily find different alliance systems that would evenly match the British side with the American side. For example, an alliance of the British, French, Italians and Japanese (plus those countries empires) versus an alliance of the USA, Germany and the USSR would be a fairly even match.

But it's hard to imagine the UK of OTL forming the sort of strong ideology or control of information that the USSR of OTL had.

As such, what would be worth it for Britain to directly oppose the US over? In OTL, the British preferred to placate the US even when a war with the US would have been a relatively cheap affair. In the 20th Century, a war with the US would be the most disastrous thing the British could do short of letting the Nazis have everything they wanted. This suggests that either the British or the US have some sort of extreme change in ideological outlook that makes a fight between the RN and the USN a worthwhile risk.

Also, British and American decision makers are going to have much better information with which to judge the relative balance of power on. So the British could easily be more formidable than the OTL USSR throughout this cold war, while also being much less frightening to Americans.

I don't think the UK has the capacity for the kind of long-term push and pull that the USSR and USA had OTL.

Arguably, the USSR didn't have the capacity for the long-term push and pull it had with the USA in OTL. The Cold War was ruinous for the Soviets in many ways.

Arguably, the USSR didn't have the capacity for the long-term push and pull it had with the USA in OTL. The Cold War was ruinous for the Soviets in many ways.
True, but that was as much about how they managed themselves in response to the US as it was about what they were simply capable of.
It's conceivable that the USSR can play its hand better in the 20th century and come out the New Millennium in a position of strength.

It seems unwise to consider OTL as a foregone conclusion. Especially when, in the end, it was the Soviets themselves who undermined their own system and geopolitical position. Not that the US didn't have a major hand is pushing along the events of 1989 and 1991, but that's blaming a match for lighting the gasoline.

I don't have that kind of faith in the British Empire over that same length of time, with the same/similar intensity of conflict. Their position is far shakier, their economy is far more dependent on maintaining control of its dependencies, and their vulnerability to military overstretch is just leagues worse than the Soviets ever had the ability to be.
It's conceivable that the USSR can play its hand better in the 20th century and come out the New Millennium in a position of strength.

True enough.

True, but that was as much about how they managed themselves in response to the US as it was about what they were simply capable of.

You can say the same thing about Britain. At any point between the start of last century and the start of the US-USSR Cold War (at which point Britain and the US being on the same side is highly unlikely to change) the UK and its empire had superior resources and population to the USSR. If we discount those parts of the empire the British are extremely likely to lose no matter what, the British empire would still be superior in economy, resources and probably population. And, as mentioned before, while Britain alone would require highly proficient management to be superior to the entire USSR in its economy, it is within the realm of plausibility.

The problem the British have is that they have a very short window of time in which to wake up to the fact that they need to treat the people in the settler colonies as equals now. To keep the non-Dominion parts of the empire, they need to come to similar realizations about the people they'd conquered with darker appearances. It is possible that a better managed WW1 or WW2 combined with leaders at the right moments who were less attached to Westminster having all the power in the empire. Potentially, Winston Churchill catching a bullet in the Boer War might be enough for some sort of imperial federation to emerge.

The problem the British have is that they have a very short window of time in which to wake up to the fact that they need to treat the people in the settler colonies as equals now. To keep the non-Dominion parts of the empire
If the Empire can somehow integrate their colonies into a functioning polity or even a strongly bound political network, then disregard everything I said.
Everything I said is dependent on the UK being just as sloppy or even worse with their colonies while attempting to combat the US. But if they can reform, then that shores up their greatest weakness.
A UK-USA 'Cold War' is possible. There was a lot of British/US tension throughout the 19th Century, and by c1900 the USA did seem to be seeking 'a place in the sun' - overseas colonies [Philippines, Cuba], building up a decent Navy, Panama Canal and so on.

However, I think the 1920s to be the most likely time for the 'Cold War' to break out.

Stroke #1: War Debt. The USA loaned the UK colossal sums to fight WW1. The UK in turn leant much of this out to her allies, such as Russia. Come the end of the war, some nations won't pay, others can't pay - London is left holding the bag, and Washington is adamant of full, prompt payment. The UK tries to get the cash out of Germany, but fails. The British people have to dig deep indeed to pay off this debt. Greedy Americans!

Both sides are left sore from this. London thinks the Americans have been unreasonable - after all, it wasn't the British who 'really' owed the money, it was the folks like the Russians. They were just the middleman! Washington thinks the British are trying to welch on their debt. Perfidious Albion!

Stroke #2: Naval Ratings. It's agreed in principle that the USN and RN would have 'parity' at the naval conference. However, the British argue that they need more cruiser tonnage to defend her large merchant fleet and Imperial holdings. When this is blackballed by the Americans, the British then try to get a separate 'Dominion' tonnage to get the same thing under a different name. The British think the Americans are being stupid by not recognising the 'special circumstances' the RN was in as the world's policeman. The Americans think it's a ploy to keep the world's oceans a 'British lake' that they could bar American trade from at will.

Stroke #3: Japan. British-Japanese relations are close; a partial military alliance, many Japanese capital ships made in the UK and so on. The Americans are pressing the Japanese 'naval quota' as low as they dared, and they're insisting the alliance is broken for the Treaty to go ahead. Tokyo feels bullied, London feels being forced to choose. In RL, she chose to submit to the American demands - but what if London went with 'heart' rather than 'head' and decided to keep the alliance with her Japanese friends and walk out of the Conference together instead?

Stroke #4: Ireland. 'Eire' is trying to escape from British domination. A civil war rages, between British and Irish, Catholics and Protestants, Moderate and Radicals. A bloody-handed 'police action' is in progress, in front of the eyes of the media. Irish American opinion is seething; they're raising much cash 'for the old country'. Like later on with the Provos, some of this 'charity' goes on guns and bombs. In RL, London decided to ignore the fact that a foreign nation was arguably funding an insurgency within the UK. But what if they had made more out of this fact? What if 'American support' had got to the level of 'volunteers' to go and fight?

Stroke #5: Tariffs. In the 1920s, the British Empire increasingly abandons Free Trade - from films to cars. However, tariffs and quotas on American-made products don't apply to American-owned subsidiaries within the Empire. What if they'd decided to count said subsidiaries as 'foreign', and thus angering American business interests?

Stroke #6: Prohibition. Canada, Bermuda and the Bahamas were making a lot of money from the alcohol trade in this period; enough for the Americans to repeatedly demand the British/Canadians to stop it. From as far as I can tell, the British simply nodded and did nothing much. But what if they'd been more forthright in their denial, making Washington look weak?

Stroke #7: Canada. Of the whole Empire, Canada was quite obviously the part most closely aligned with the USA. In fact, the first 'foreign diplomatic action' she did in her own name was a trade treaty with the USA [In RL, Washington asked London first if it was all legit]. However, what if London had striven harder to keep the Canadians within the 'British' orbit, thus making a Canadian-American treaty impossible?

So, there's enough fuel to make the fire; it's only that both sides decided not to make one. But... what would said fire look?

First off, I think the USA would be more anti-colonialist than in RL; she would play on being a 'democratic nation' in comparison to the 'oligarchic' British. This means there could be some Soviet-American cooperation on this front at least. Said anti-colonialism rhetoric would drive the European colonial powers to support London. The Germans more likely to become pro-American - from a combination of hating the Anglo-French and American loans.

I foresee two major 'theatres' of rivalry; South America and East Asia.

South America contained much British capital/imports, and many nations are ruled by local oligarchies. If Washington can buy off said oligarchies or 'remove' them, they can then 'tilt' them towards accepting US capital and imports instead - which might be vital if much of the world is 'locked out' by the British. The British, on the other hand may think it wise to try to strengthen the likes of Mexico in their quest not to end up a de facto American economic colony - which Washington would surely see as 'meddling in our backyard'.

East Asia would interest everyone - particularly China. It's quite possible there would be a situation where the Japanese are trying to increase their Chinese 'holdings' with British tacit approval as they bribe Warlords, while the Americans preach freedom and prosperity while bankrolling the Nationalists. American 'agitation' might be seen throughout the region; promising rapid economic development and a lack of racism to people where 'European' is usually a variant of a pith-helmeted scoundrel reeking of gin ordering 'natives' to be thrashed because they didn't harvest enough teak today. However, this depends on what the Americans 'do' with the Philippines; all the propaganda in the world will mean little if the Americans in Manilla are acting almost identically to the Dutch in Batavia and the British in Kuala Lumpur.

The one thing I doubt is this; that it turns into a massive shooting war. The occasional 'incident' on the high seas, yes. Proxy wars, perhaps. But an American - British war would ultimately end with a stalemate perhaps somewhat in London's favour; while the Empire would lose Canada, it's likely she'd gain the Philippines and Guam, have successfully damaged American trade and if it had gone 'well' [such as having Japan a full ally] might have even made a play for the Panama canal, Puerto Rico and Hawaii.

This was the traditional British tactics in war. Focus on your navy, trade lanes and taking your enemy's colonial holdings. Then you sit tight and then make a compromise peace [retaining a colony or three] when the enemy grows too tired to continue.
In your scenario, @KarmaPolice who do you see 'winning out,' for however long it may be? I think per the economic situation, Britain would probably be in by far the better position- after all, they still control the trade lanes better ttl with their navy and are allies with the other prominent Boat People, and can probably afford some protectioniism to properly damage America's power. Unless America gets desperate enough to align with a certain other jipped power britain won't let have a navy?
Unless there's a 'surprise' [such as a Japanese/French entry] I personally think it would end with a stalemate. Any direct USN-RN battle would likely be a repeat of Jutland; lots of damage, few direct sinkings. The USA would be suffering economic distress due to the blockade, but the British can't keep up a war of this level for long. The Americans will win the 'war on tonnage' eventually. It's possible that while the Philippines, China, Guam, Hawaii and the Asian/European trade is 'lost', Canada and the S. American trade is 'gained' and Panama, Cuba and Peuto Rico is 'held'.

Personally, think both sides would grope back to some status quo, once the 'true enemy' appears in both homelands - the spectre of a Bolshevik revolution.

Germany wouldn't offer anything at this period, and while the Soviets would try to take advantage it would be to incite the very above...