Does Britain really care what America is doing? Honestly, one of the most plausible and interesting parts of LTTW is the Empire of America, which is part of Britain but basically does its own thing most of the time.
The problem is simple: the 19th century was a key period for the US and the UK. They can't just do their thing ignoring completely what happens in the other side of the Empire does. It's just too much: Manifest Destiny, Mexican-American War, Crimean War, consolidation of British India, Suez, colonization of Australia, South Africa, etc.
EDIT: Also, as I pointed out, tax policy is the elephant in the room in this union.