I don’t think it would have made much difference in the early 1800s with Napolean, the USA was not really a source of men or anything else. It was just a large underpopulated land mass rather a long way away.
Canada and Australia rubbed along quite happily with us, perhaps the ‘them’ and ‘us’ stuff was less extreme - I’m not really sure.
The USA would definitely /not/ have had the Civil War, we abolished trafficking in slaves in 1807, and I can see that steam powered machinery for harvesting cotton etc would have been piling out of the UK. Incidentally I read that the USA was a nett importer of agricultural equipment in 1914 and a nett exporter in 1918 - presumably canny engineers migrated in droves.
The big difference would have been WWI - by then the USA would have been a really useful source of food, men and munitions - Wilhelm would have thought twice if /knew/ that he was taking on the USA.
Rudyard Kipling would not have been born in the USA, he was actually part Indian, he spent an astonishingly small part of his life in India, his story telling abilities probably came from his Indian Amah (nurse). She, or perhaps a Syce (groom) must have been very talented.
There is a massive difference between ‘colonies’ and ‘administrative regions’, India and most other British dominions were not colonies. India was a free enterprize operation where John Company hijacked a sub-continent, ran it rather badly so we had to nationalize it.
I think it is possible that we lost America because we sent out Prussian (or was it Hessian) mercenaries instead of British troops, I heard, but cannot cite, that it was a very real possibility that the official language of the USA could have been German. Perhaps someone knows more about that.
As a WAG, it is possible that the UK government might have migrated to America.
Immigration would have been the same, the UK has a long history of immigration and America was something of a vacuum.
I guess that if my and your British ancestors (which you probably have, regardless of family history) had been a bit smarter, America would have industrialized agriculture earlier, abolished slavery much earlier, the Civil War would not have happened, and WWI would not have happened - and that means that Hitler would have not been able to feed on the resentment of a defeated nation.
It is also likely that British administered areas like India, Egypt and swathes of Africa would not have seen their thinly spread administrators as representatives of a small, beleaguered nation that is barely visible on a map, but as the representatives of a monstrously large and culturally homogenous land mass that was worth keeping on good terms with.
I’ve not really thought this one over much before, but it looks as if there was a bit of a cockup.