Just some random thoughts on settlement of the BSA:
IOTL, the U.S. South didn't see much immigration between the Revolutionary War and the modern era except in a few localized areas (Texas Hill Country and New Orleans mostly). Presumably, IITL there will thus be more migration into the DSA than our history saw. Still, it needs to be kept in mind that the climate is, while livable, not exactly the first choice for migrants from Northern Europe. Also, even with the end of the slaveocracy, sharecropping will probably remain the norm in many regions, meaning less opportunities for advancement for someone "fresh off the boat."
The question is, who is going to come?
While IOTL English, Welsh, and Scottish migrants mainly came to north, presumably Britain will be pushing them a bit more actively to come to the DSA (particularly to settle in formerly rebellious places). I'm not sure if the DSA will even be able to meet the immigration "pull" of the USA, but at least it could keep competitive. Say perhaps a 60/40 split in favor of the U.S. It would be interesting to see a larger Welsh population somewhere in North America - I think Britain's share of the Ozarks would be a prime settlement area, although the majority of the Ozarks are still in the U.S., and at least in the early days, the border there will be porous, meaning both sides will probably be fairly culturally similar.
What about the Irish? My gut instinct is they will still travel heavily to the U.S. Both states are somewhat Catholic-friendly now (USA has Quebec, DSA has/will have large Francophone and Hispanophone populations). Still, not only will the economic opportunities still be greater, there will be less of a desire to migrate to a British-ruled region than for Welsh, Scots, or English. You'd probably see something like a three to one ratio in favor of the USA.
The major place outside of Britain I see the DSA turning for migrants is Portugal, for a number of reasons. First, the two nations were historically close allies. Secondly, there was a fairly substantial migration (at least, substantial as European migration to the region goes) of Portuguese to some of Britain's Caribbean possessions IOTL. Thirdly, IOTL Brazil had a huge "second burst" of Portuguese migration, which ran from 1881 to 1960, but was greatest in the early 20th century. With Brazil looking to be somewhat in shambles, some of those people will be looking for other opportunities.
I'd also expect, assuming OTL's close relationship between the UK and Greece remains intact, significant Greek migration.
One unanswered question in all of this is will Hispano-America keep its shit together more than IOTL? If so, it's highly possible that one or more of the countries will be more of a pull for migrants, particularly from Catholic nations, and that the Irish, Italian, Portuguese, and Polish populations for the USA and the DSA both will be somewhat reduced.