I'm going to go out on a limb here and say Britain and the colonies came to a peaceful agreement - maybe the Olive Branch Petition actually works - granting the colonies individual *Responsible Government, then a dominion-style thing happening around the same time as the Constitution/Bill of Rights/Canada Act (1787/1791) since the concept of a united America was around since the Albany Plan (1754).
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There will be some reactionary war in the 1770s-1780s by the Bourbons against British waxing of power, but here the Anglo-Americans are united. It'll likely be a stalemated war with some minor Bourbon gains in the Caribbean due to North America's mainland being now-impregnable with a united Anglo front. France's money scheme with or without an American Revolution or some sort of revenge-war in this timeframe will still cause the French Revolution to happen, give or take a couple years.
Louisiana would definitely fall during the French Revolutionary Wars both from demographic expansion but also specific military expeditions. However, northern New Spain - Texas, New Mexico, Upper California - would as well, for the same reasons. Anglos had already moved into Texas in small numbers and revolted as the Magee-Gutierrez Revolt proved, and a singular ship captured San Diego during the later period of the aforementioned war. The difference being that these expeditions by American filibusters will have the British Empire, including the American dominion, very likely supporting them in TTL. And America both colonial and independent focused entirely on North America straight up to 1898 in OTL, there's no reason for it not to be the same as a dominion, handling North America while Britain and its navy deals with Napoleon up front. Americans will slowly take control of domestic affairs during the fifteen years or so of warfare of the FR-Napoleonic Wars and gradually gain full independence in the coming decades as Canada and Australia did, just 50-80 years more early than them (IE, 1840s-1860s vs 1920s-1940s), but still remain in a firm alliance with the mother country.
Louisiana and Texas will cause the British a ton of headaches to settle the issues their subjects in America raise.
British North America will inevitably divide into several countries, so probably Maryland and south will form one "Confederation", New England (maybe with the Maritimes) will form one, and Pennsylvania, New Jersey, Delaware, and New York will form the third. Not sure what might happen to the Canadians in this case, but we can assume that Canada as we know it won't exist. However, the concerns of British North America will increasingly dominate the British Empire. Slavery will abolished much later in the British Empire, for instance.
Well, Canadians moving west caused headaches with the Metis and local Amerindians.... but expansion still happened, so it will with a united Anglo-America.
Why is British America splitting up inevitable? Nobody ever ACTUALLY says why, just "because". Meanwhile, the concept of uniting the American colonies has been seriously considered since 1754 and the Continental Congress in both First and Second forms invited the Floridas, Canada, and the Maritimes to join it constantly. Even British North America proper - IE, Canada as we know it - had a number of unification schemes that involved all of the colonies at the time. To say nothing of all Australia uniting and expecting New Zealand to join in, even Fiji coming along for the ride. Heck, South Africa and the West Indies Federation, as well! British settler colonies being EXPECTED to unite as one big federation is the norm, even if certain colonies don't join up (IE, PEI with Canada at first, New Zealand vis-a-vis Australia, etc.).
I think northern New Spain still would have been taken if they allowed western expansion eventually they would have had wars with them and taken it as concessions Cuba and PR too. Here the Caribbean probably would too have eventually ended up British too. In this would the sun would truly never set on their empire London would the capital of the largest most democratic nation on Earth and they'd have toes in everyone's empire too.
All of North America probably would be theirs, and some of South America too.
They'd have large continental holdings in Europe too.
Britain already had the lesson of the ARW when it took over too much too fast, they had all of Europe against them openly or subtly (League of Armed Neutrality). There'd be a ton of concessions to the rival empires and returning of colonies. They already had enough land to settle in the *western USA and *western Canada alone.
Tons. Spain would never rest if it lost Cuba and Puerto Rico.