Supposing that ARW doesn't happen, that perhaps the colonists don't come to blow with the Brits... what happens once the colonials go west, and North America becomes a far more populous and prosperous region than Great Britain itself. Does government and nobility and big business establish itself in a new capital in the United States? Or is this great commonwealth resigned to being ruled from Europe, until such time independence by way of war or dominion come to pass?
It will ultimately depend on just what happens after said POD, but it seems to be most likely that there will probably be something not far removed from the C.N.A. from
For Want of a Nail taking the place of the colonies, and whatever comes under British rule after them.
However, though, I'm afraid that Sobel's rather hopeful & optimistic vision that he presented to the world, isn't quite all that realistic in some areas(an African-American governor in Southern Vandalia.....in
1914?).
Here's an idea of what to look for:
1. Sadly, it's rather unlikely that slavery will end in 1833, as it did in Britain's Caribbean colonies IOTL. There are a variety of reasons for this, but one important one is that, without the 13 Colonies, Britain lost a huge amount of future potential profits that they might have gained from cotton, etc.; but also, perhaps just as importantly, the victory of the Patriots in the Revolution was ultimately seen as a victory for the ideals in which they upheld(or tried to!), something that the abolitionists and other liberals(not quite in the modern sense, though) of their day took, and ran with. If you want a realistic estimate, I would go for probably somewhere between 1860 and 1880 for the beginning of the complete and total end of slavery in *British America. Anytime before 1860 is pushing it a bit, but after 1880 is also unlikely.
2. Mexico might be left alone, by and large, unless a *Santa Anna type figure comes about.
3. Canada may or may not be absorbed into *British America.
4. I honestly don't see the monarchy wanting to leave Britain for the Americas. Chances are, we may actually see a system similar to OTL Canada(complete with Governor-Generals)