The United States of America would not exist, full stop. The elites behind the (first) American War of Independence would not be given the chance to start a second one any time soon if the British are even halfway competent at colonialism, which they definitely are. They also probably wouldn't even be able to get all the colonial governments to work together again anyways if they did somehow manage to escape any consequences from the first war and initiate a second. Honestly, if the actual OTL American War of Independence was part of a timeline, a lot of people would call it ASB or at least authorial fiat. Add in the conquest of the majority of North America's best lands by the USA, without much contest except from Mexico and with France practically handing its colonies over on a silver platter, and it'd definitely be considered ASB. The chances of a second war happening are slim, and the chances of it succeeding even more so. The idea of the United States all seceding together as one country, and then staying that way, would appear as unworkable in TTL as the idea of every Hispano-American country merging into one.
There'd still be post-colonial Anglo states in British North America, which would probably include all or at least most territory east of the Mississippi eventually, but it's very unlikely these would merge into two huge countries with the territory of OTL Canada and the USA. You're much more likely to see multiple smaller states, most or all of them part of the Commonwealth of Nations or some equivalent. I expect New England and the Maritimes would be one country, for example, despite the former being American and the latter Canadian in OTL. The lack of a threat from the USA being independent means the region's cultural ties won't be severed, but will instead grow even stronger. Similarly, the people living in areas around the Great Lakes will likely regard each other as kin more than they will the coastal regions. The South might remain as a recognizable politico-cultural block, though perhaps split between the Upper South and Lower South or between Appalachia and the lowland areas, and might revolt over the British Empire banning slavery. The Middle Colonies are probably considered one coherent cultural area as well.
I doubt any Louisiana Purchase happens, or that the British would care as much about territory west of the Mississippi, but I also doubt the French would colonize much beyond the mouth of the Mississippi due to their scarcity of colonists. I would, therefore, expect a significantly larger number of Métis living west of the Mississippi than OTL due to prolonged native contact with the French. As such, various Métis/Native countries are likely to exist, though probably under the thumb of one great power or another. California, New Mexico, and Texas might remain Mexican, as well. Butterflies may change things entirely to such an extent that nothing recognizable occurs, but just saying "butterflies change everything" is boring.
I expect presidential systems would have nowhere near their OTL popularity, with parliamentary systems being far more dominant. Additionally, constitutional monarchy might be more common in North America, though probably mostly under the British Monarch like OTL Canada and Australia. Possibly some sort of Organization of Anglo-American States forms in the future, or maybe an EU equivalent, but that's too far away from the PoD to talk about with any detail.