Maybe if you combine a massive Britwank with a Yankscrew.
Britain remains neutral in WW1, whilst America gets dragged in earlier on France's side. American intervention is enough to keep France and Russia fighting longer, but isn't enough to swing the war in their favour. The Central Powers win and America now has an even larger number of pissed off veterans upset that they were sent half-way round the world to fight in a pointless war.
Britain, on the other hand, is able to sit back and grow rich by trading with both sides and learn about the new developments in modern warfare from a safe distance. After the war Italy and France have Communist/Syndicalist revolutions, which Britain (along with Germany) uses as an opportunity to seize some of their remaining colonies.
During the 20s things go downhill for America domestically with rising political violence and the KKK gradually taking control of the Democratic Party (see Ruins of an American Party System but without the Democrats completely collapsing).
A great depression breaks around about IOTL, but the American political elite are completely incapable of responding to it (think of a mashup between Ruins of an American Party System and The Falcon Cannot Hear), eventually resulting in the outbreak of a devastating Civil War. Britain, by contrast, is able to implement a successful Keynesian program which alleviates the worst of it.
America gets torn apart in a massive multi-faction civil war for a few years, completely destroying faith in the old union, until British intervention helps bring it to a close. the US is divided into multiple states, several of which become British dominions. Over the following decades Britain gradually extends its sphere of control over the former US, eventually unifying the English-speaking parts of the Americas into an American Commonwealth under the British crown.