So I've seen a couple of threads discussing this type of scenario, but I'm specifically curious how this would affect the development of the young United States.
So suppose that for whatever reason the British just do better in the southern front of the Revolutionary War and by the end of things they hold on to what would have been considered the US Deep South in OTL- Georgia, South Carolina and East Florida.
Borders end up looking something like this:
So I'll reiterate, how does this likely affect the development of the young United States. It's government, expansionist ambitions, presumably more northern dominated politics, and of course absence of a lot of the slave states.
Not having control of the east bank of the Mississippi River south of Memphis could have a big impact on westward expansion of the US. Presumably, rights of navigation would be an issue that would be covered in the treaty ending the war, but there's still a certain amount of awkwardness involved when the river is entirely within foreign territory, and forms a boundary between two different foreign empires at that.
If the British hold on to East Florida in 1783, I'm not convinced they would cede West Florida back to Spain, but obviously we don't really know what the rest of the war looked like and how it might have changed due to Great Britain doing better in GA/SC, so maybe. Any of the following could be plausible outcomes:
a) West Florida remains British in 1783 along with East Florida (in exchange for concessions in other parts of the world? Belize, perhaps?)
b) West Florida returns to Spanish rule per the quoted map and Spain keeps it for an indeterminate period of time.
c) WF returns to Spanish rule in 1783, but is later acquired by Great Britain, the US, or split between the two.
Either westward expansion of the US is greatly curtailed, or access to the Gulf of Mexico via the lower Mississippi will be a high priority. If, for example, Americans who wish to ship goods via the Ohio/Mississippi are compelled to sell their goods to Spanish or British middlemen, that would likely reduce the appeal of developing the Ohio valley. I can't imagine that the US has the resources to buy Louisiana much earlier than OTL, and I'm not convinced it'd be for sale earlier anyway, so westward expansion may be significantly slowed.
Assuming that despite all butterflies in the meantime Louisiana comes up for sale at some point in the 1800-1810 period, does Great Britain decide they want it bad enough to take it by force (maybe not unreasonable if they're at war with whomever happens to control it at the time) or outbid whatever the Americans can pay? British Louisiana probably constrains the US to their c.1799 borders.
If Great Britain either decides they don't want it or decides that it's safer in American hands, this still does not give the US full control of the Mississippi River like it did in OTL. Certainly Georgia having the OTL state of Mississippi means that some agreement must be made between the US and Britain concerning navigation rights on the river, even if the US gains control of the west bank of the river.
Regardless of what happens with regard to Louisiana, without the OTL states in British Georgia, there should never be a situation where slave states can successfully block admission of free states. In OTL, Alabama (#22) was the last state admitted before the Missouri Compromise of 1820. In TTL, Alabama -- along with Mississippi, Georgia, and South Carolina -- is still British so there are only 18 states (17 if Louisiana doesn't happen), of which only 7 (or 6) are slave states. Slavery probably pretty much dies out in the US by about 1840 or so, cotton gin or no cotton gin.
I am uncertain how any of this affects Texas. Could be anything from "basically just like OTL except a significant percentage of the slave-holding Anglo settlers are British subjects" to almost no Anglo settlement in Texas. Either way, I think there's a pretty good chance that Texas never becomes American. And I'm really not going to project beyond 1850ish, just because the accumulation of 70ish years of butterflies means I could plausibly justify a whole bunch of different outcomes.