I'd say - Civil War that somehow (perhaps French or Spanish intervention) leads to a Confederate victory, or the Civil War is going against the Union, significantly. I would have to say that a couple of countries would have to be backing the Confederacy. France might be able to do it, if they were less cautious about British involvement - and that could be spun well in the Confederacy. "France has always been the friend of an Independent America".
Britain is approached by a rogue politician who asks for British protection, in exchange the "Union" becomes part of the British Empire. This then leads to a serious debate leading to the Union of British American States (or some other entity). Instantly the British intervene with their own blockade and preventing French assistance (in Europe the war is mostly cold, France being unwilling to suffer large losses for the Confederacy). With the Union as a British Imperial Dominion, the Confederacy is pushed to a stalemate, making a peace treaty to be independent OR to control the lands West of the Appalachians. Anyone pro-slavery or Anti-British migrates over the mountains, and leaves the larger part of the population in the UBAS in the British Empire.
The only State that is independent in America is much smaller - and centred (most likely) in Texas/Louisiana, but still grows because it has vast plains to exploit and farm - but not very densely populated. Britain dominates the Lakes and the Union, and continues to undermine the Confederacy by opposing slavery and isolating them economically. Limited economic and demographic increases leave them a large state that tries to encourage other communities to declare independence from Britain and France, promising a deeply decentralised government. It doesn't succeed massively in the first 70 years, but does so later - after giving up slavery due to the British embargo.
