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.
