That's still a lot of time.

You have the rebellion itself, but that's over by the mid 1780s. There's definitely still a lot of tension though and these come to a climax in 1812. This is because just because the war ended it doesn't mean all the involvement did, and all the money and investments went away. There's still a vested interest.

The Monroe Doctrine in the 1820s is the official notice of "stay away" I think this is the point to draw the line.

Now think about what happens afterwards in both countries. Think about what the Monroe Doctrine represents. It's not "America protecting its allies in the continent" it's "America claiming these territories as their own sphere of influence" England still has the other half of the world to push its own imperialist tendencies. We're no longer colonising, but we're still reaping the benefits out of every place we can find and bringing them home.

And sure, there are some disputes after this, but overall they're not really the end of days type stuff. It's a lot of chest bumping more than sword shaking. Even with the Second US Rebellion in the 1860s the British considered getting involved just to tip the scales a bit in their favour, but overall realised that diplomacy would win the day, and it was better to just sit back and wait and act accordingly afterwards.

This sparked a more open and diplomatic approach which had already been growing. Better to work things out, even we have to act a bit tough for votes, than have young people go die fighting each other. Especially as both were now monumental powers. It's easier to ally with them and use each others strengths and say "hey, look, we scuffled, but we're both from the same cloth" by this point people that fought in the American rebellion or the war of 1812 are no longer alive, or certainly not the majority of the population, there's not that same animosity. "You scratch my back, and help secure trade in the region, which we'll help make you a part of, I scratch your back and I help ensure a peaceful Europe beneficial to you"

So when WWI approaches, the US realised how important it was to keep the Brits as an ally, and to keep them powerful, if it meant a safer Atlantic for their own interest, especially against growing nationalist groups (in particular Germany). The actions of WWI consolidated the US's thoughts on Germany and similar groups, and shone the importants of an ally in Britain. After the war they had to work together to try and rebuild, strengthening them more and realising again the need for continued cooperation, which would continue to prove itself useful.

In short: It's easy to look back at big events of centuries and go "why aren't they hating each other?" but in reality, people care more about what's happening NOW than what happened before they were born. Yes, there are certainly groups which harbour some amazingly longstanding feuds, but even a lot of those still have some relatively diplomatic relationships. A lot can happen in a few generations.

Indeed it took a good deal less than a century: only a decade after the 1812 war London was proposing a joint declaration against Europe's monarchies seizing territory in the Americas; Monroe chose to go it alone, but Britain backed the policy despite implicitly being among those being warned off. Trade shows a still faster recovery in the 1780s and 1810s: within a few years of each war you'd think nothing had happened.

Besides the obvious affinities between their ruling elites, part of the explanation is that with the US renouncing any involvement in Europe's affairs and preoccupied with westward expansion across the North American continent, there was for a century little basis for friction once British rue was gone, apart from the 1840s border dispute, Civil War complications and the brief Venezuela flurry. The two powers shared a distaste for European rivals' imperial designs, Washington wanting to keep Europe "over there" while London prized commercial access to non-European lands, a growing US priority too by mid-century: nor until the 1940s did the US show any inclination to assume the global financial leadership claimed by Britain.

It was ultimately a marriage of convenience, based in the first century on broadly compatible strategic and commercial perspectives, and in the second on waning British capacity to go it alone as others challenged its industrial lead and - unthinkably - its naval supremacy. Once Britain abandoned any fantasies of reconquering its lost North American colonies and US hotheads were talked out seizing Canada, there was little to do but make the best of it.

Please don't underestimate the immense value of a common language (despite G.B. Shaw's famous quotation!) in the hallways of politics and diplomacy. Nuance and idiom are better understood when both parties are speaking their native language. In the U.S., Americans, who are famously and woefully uneducated in any other languages (over 90% of native-born white Americans speak only one language with any fluency), are forced, as were their ancestors, to seek rapport with those who also speak their language. In international diplomacy, there is a perennial distrust of those with whom one cannot smoothly converse. Famously, interpreters in one-on-one conversations will pose a yes-or-no question on behalf of a speaker, to the other individual. And after perhaps a 30-second or more interaction, will turn to the questioner and say, "He says 'yes.'" Well, no, of course he didn't say "yes." He sought clarification, or posed a different response subjected to condensation by the interpreter. Not uncommon.

So, the Americans and their UK cousins, working around accents and idioms, were able to carry on much smoother and understandable deliberations and compromises, than either group could accomplish with any other nations (outside the Commonwealth, who unsurprisingly also remain our closest allies). Once again, famously, the 8000+ mile border between the U.S. and Canada, which has gone unguarded for over two centuries, is unprecedented in world history.