"For Want of A Nail" by Robert Sobel is the classic AH account of if the British had won the Revolutionary War. Its a must read, IMHO, as to whether or not its ASB is another matter.
In a nutshell, Washington is defeated at Saratoga. There are a few more battles after this but eventually moderates come to power in the Continental Congress and sue for peace.The victorious British are initially harsh and vindictive. Washington, Adams, Paine, etc are hung for treason in London. Jefferson lives out his life in exile in France.
There is no French Revolution or Napoleonic Wars. The decadent European aristocracy remain in power until the 1870s or so.
The British after much debate eventually grant the American colonies a limited and complicated form of home rule, the "Brittanic Design" with John Burgoyne as first Governor General of the United American Provinces. Disaffected rebels immigrate to our Texas, nominally controlled by Spain and then Mexico and form the Free State of Jefferson.Cotton grows well there and after much debate Jefferson agrees to allow slavery.
In the 1820s Jefferson intervenes in a civil war in Mexico and under Andrew Jackson takes over that country and forms the United States of Mexico.The USM aggressively pursues its own form of Manifest Destiny. From the 1850s through 1860s there are a series of wars,the Rocky Mountain War, between the USM and the UAP for control of, well, the Rocky Mountains.The British fail to adequately aid the UAP which leads to much anger and bitterness.
Slavery is peacefully abolished in the UAP in the 1840s.
In the 1880s, occur the "Bloody 80s" a complicated series of revolutions in Europe. France has a communist government for a time.
In the early 20th century after much labor unrest the UAP gradually becomes a Scandinavian style social democracy while slavery continues in the USM until the 1920s.....
Anyway I think this is plausible up until the 1830s or so. I could see Texas or the southern Louisiana territory being colonized by diehard rebels. Whether they could defeat, no less annex Mexico is another matter.A seperate economic power may develop in our Texas. Louisiana Territory and our Mexican Cession may be a contested region. I don't think France, under any regime, would be able to or really much interested in holding on to Louisiana.
My guesses...
The defeated colonies would probably be loosely united in 2 or 3 weak confederations but would remain closely linked to Britain. Quebec would be larger and become independent earlier.
Slavery in the South might collapse earlier.
An independent "Texas" might rapidly assimilate California and much of OTL Southwest. My guess this state would be chaotic,corrupt and unstable."Texas" might form a pole of attraction for slave holding southerners in the 1840s. Its difficult to tell how things might play out.There might be a series of wars between "America" and "Texas" for control of real estate. These struggleswould not be as principled as the US civil war but wouldbe seen more as tawdry business conflicts.