Ok, so lets say that the US was not allowed to be settled by heretics, and that the king gave out land to aristocrats. Basically, an all Anglican society with English landholders, black slaves, and white tenant craftspeople and servants, where colonial governors are directly appointed by the British crown (at least until the revolution, in which the aristocrats rile up the masses to take power for themselves).
It's not exactly the most likely path, but let's take the premise as given. I think we may safely assume that if geopolitical developments (meaning: the removal of the French threat) go 'on schedule', so will the American uprising against British rule. So there's that, making things a bit easier to compare to OTL.
That said,
@Dolan is right. Don't expect any kind of Republic. These aristocrats are going to be looking for a king. They may put the most esteemed of their own number on the throne, they may get a sympathetic member of Royal Family to become their monarch, or they may invite a foreign candidate. Granted, considering the spirit of the times (if this is anything like OTL), it's possible that they implement an elected monarchy, with the most prominent aristocrats of North America serving as Electors.
How does the US expand? Does it Buy Louisiana? Take Florida (or maybe just part of it, to complete Georgia's promised territory)?
The thing is, if this model of different colonialism is chosen, it's not somehow going to be applied to just the thirteen colonies. It's going to be applied in the rest of British North America as well. Since this kind of social hierarchy and governmental structure would be less conductive to the ascent of a middle class, you're going to see the landed gentry as the main factor of importance across the board. If colonial governance is objectionable to that class (as it must be, to get a revolution/secession going) then it's going to be objectionable to them all over the place.
Therefore, you'll probably see
all of British North America split off. That changes the way further expansion is handled. I suspect that 'clearing out' and settling the Ohio Country would be the first priority. Further wars and/or purchases depend largely on international relations. However, I strongly feel that any country taking the place of the USA will inherently share the USA's OTl desire to possess New Orleans. As long as that's in foreign hands, whoever holds it can strange off a
lot of American trade. So that'll surely become an issue they'll have to deal with.
Do Virginia, North Carolina, and Georgia stay huge (since they're super influential, TTL, they would control the government, and thus not let themselves be carved up, right?) Does New York keep Vermont?
One may assume that popular soveignty is less important. Big states will remain big, and can just be internally divided into lesser fiefs. (For instance: the County of Vermont, which is part of the Duchy of New York.)
I agree that Virginia is likely to be the heart of this America, which may cause issues if -- as I suspect -- it extends far to the North. Certain Northern regions are of least interest to the landed gentry, and are most inclined towards a half-way prominent bourgeoisie. A centre of republican unrest? Possible! And Quebec may well chafe under American rule, especially if this America is intolerantly Anglican. (They won't be naturally inclined to respect Catholic rights.)
Industrialization and abolition almost certainly happen later than OTL, when would this happen roughly? Would it be as late as Brazil, or a little earlier, around when most South American countries abolished Slavery?
Keep in mind that an aristocratically led nation must retain the respect and confidence of its European counterparts. Becoming a pariah is unacceptable. Abolition may ultimately end up being something they can't avoid. It'll be the aristocrats who own the slaves, and they'll be the ones suffering when they become seen as brutish foreigners in Europe. When it's a choice between taking an economic hit and forever saying goodbye to respectable European marriage prospects... you clench your teeth and take the economic hit.
I suspect that this means that abolition will happen around the time it did in Brazil. There will be no Civil War, I think. If such a conflict arises, it'll be about a secessionist republican movement in the North (meaning New England and Canada, not the North in the OTL Civil War).
Industrialisation will be slower to start, but keep in mind that in many European countries, it was funded by aristocrats looking to make a nice profit. I expect the same here. Aristocratic families will have branches all over the country. They'll own Southern plantations that produce raw materials, and Northern factories that process those materials.
A weird question: Virginia controlled Bermuda for a while OTL, would it keep it TTL (post-revolution)?
Holding on to insular possessions will be much easier for Britain than holding on to a revolting North America. Maybe Virginia gets to keep it in the peace agreement after the revolution? Otherwise, Britain just keeps it.
Would the US ever rebuild relations with Britain like it did OTL? or would it hate it like south America hates Spain (Britain TTL would be a worse colonial "parent" than OTL, but still not as bad as Spain)
There is little reason to keep hating once independence is achieved. Trade interests led to reconciliation in OTL. It would've happened sooner without Jeffersonian anti-aristocratic (and pro-French, anti-British) fervour in the USA. In this ATL, the familial connections of the American aristocrats to their British relatives is another factor that encourages reconciliation. I'd wager that by 1800, America and Britain are on normal terms again.