As our former overlords go to the polls today, a fun fact: the first time a UK government fell on a vote of no confidence was when Parliament voted to end the war with America, causing Lord North and the Cabinet to resign. It was an important milestone for British democracy, ironically enough.
Sometimes you'll see the usual debate over if independence was actually good and it will be praising stuff about the Westminster system or 19th C. UK history, and it's things that only developed after, in some cases directly in response to the Revolution. It's all heavily intertwined and contingent.
Would the UK have abolished slavery a few decades earlier as it did in real life, before some presumably gradual American independence on the Canadian-style dominion track? Maybe, but South Africa and Rhodesia don't inspire much confidence on that score.
But the (white, of course) dominions, starting with Canada setting the template, developed in that direction in no small part in response to American independence and influenced by Britain's determination to never spark that kind of rebellion again. So even that can't be taken for granted.
Canadian federalism and even more strongly the Australian constitution were also heavily influenced (for better or for worse) by the American model. They took some parts of it and not others, to be sure. But they were definitely looking over our shoulders a bit on some of that homework.
In the immortal words of our greatest president, "not just an American holiday." There's some truth to that! Especially for our closest cousins who are still hung up on that whole "king" thing.
