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.