For whatever reason, say America does worst in the War of 1812, leading to the British (Who never saw the Louisiana Purchase as legal.) to take Louisiana
Man, the British would really need to curbstomp the US for this to happen. Battle of New Orleans (or analog) would have to be convincingly won by Brits and lead to British wins at other towns in the lower Mississippi region. American outposts such as Detroit, Fort Dearborn, and Fort Wayne would need to fall and be held by the British/natives. The Great Lakes would need to be in the firm control of the Royal Navy (i.e. Perry's victory at Put-in-Bay doesn't happen). Forts on the Niagara River need to be defended effectively. The Chesapeake Bay, Lake Champlain corridor, and coastal Maine need to be captured, key towns/cities occupied, pulling US troops out of the west. New England discontent needs to be taken more seriously ITTL. Britain has to be clearly willing at Ghent to devote increasing troops to North America as Napoleonic Wars wind down. &ct. &ct. Phew!
Still, fun POD.
2: Everything of Louisiana from 42°N and above is attaching to British Canada while spinning the rest off as a pro-British state or a separate British colony run from New Orleans.
I don't see the British dividing the Mississippi watershed north of 42°N. I'd expect that either a) the whole of British Louisiana territory would be governed from New Orleans, or b) it would shortly be divided into two territories, "Lower Louisiana" governed from N.O. and "Upper Louisiana"/"Missouri" from St. Louis. The colonies would be linked to Upper Canada via trade across Lake Superior, but governed and developed as separate entities. Question is whether the British also wrest a corridor to Lake Superior out of the US from Wisconsin Territory (i.e. modern northern Minnesota). Technically the territory does not touch the Great Lakes without additional land cession, so without a connection to the lakes the new acquisition would not be as unifying as it might seem.
Lower Louisiana would come with a fair amount of development, population, and infrastructure, certainly compared with Upper Canada. Sault Ste. Marie would likely become an important trading post connecting British North America from Canada to Louisiana across Lake Superior. Depending on any additional land cessions from the US, a British Grand Portage or Duluth analog would bookend trade in the region with a portage connection to the Mississippi - a canal would probably be built eventually in the 1820s depending on Anglo-Ojibwe relations.
Westward exploration/expansion: The British would go about it *very* differently from the United States. The native fur trade/"mountain man" era would dominate. Trade outposts would develop along the Missouri River and tributaries to the Rocky Mountains, but the details would be dependent on the dominant regional native nations. The British would not be focused on opening up land for settlement for many decades.
Gulf Coast Foreign Relations: One question is whether the terms of the Treaty of Ghent gives joint navigation rights to the US on the Mississippi River. Navigation rights would make near-future Anglo-American conflict much less likely. Likewise questions of the ownership of West Florida would set the scene for relations going forward. West Florida was claimed by an "Independent Republic", Spain, and the United States (via their LA Purchase claims). If Britain accepts US claims to West Florida in the Treaty of Ghent, a source of future conflict could be avoided.
New Orleans is wedged between West Florida and New Spain/Mexico. US designs on East Florida will only increase after such a drubbing by the British and the lost of westward opportunities. That said, the US may not be in the position to claim territory from anyone in the near term, and the potential for conflict between the British and Americans on the Gulf Coast could lead the Florida can to be kicked further down the road. British goals in the Gulf could be advanced by strategic diplomacy with Spain during the Mexican War of Independence. Some sort of arrangement could be worked out over Florida if the British stay out of the Mexico conflict.
US Butterflies: Federalists back in vogue? Republican model of land expansion discredited w/o Louisiana. Focus on repairing trade ties with Great Britain, led by New England. Internal improvements back on the docket? Does Jackson survive the war? Questions about natives who fought w/ British in 1812. Does the treaty provide for their protection w/i the United States, or do the British offer them haven? Lack of land for removals sets stage for dramatically different possibilities for natives in Great Lakes region & the South. Lack of land for expansion changes the slavery calculus. Slave prices don't stay high without room for slavery to expand. Too much up in the air to say more than that. Probably will be more conflict with Britain at some point IITL's future. Regardless, US history and development becomes much more intertwined with/dependent upon Great Britain.
British Butterflies: The Acquisition of Louisiana completely changes the calculus of British empire-building. Britain is suddenly put into close contact with Spain again as an imperial power. British policies with an independent Mexico/Latin America may greatly change development in those regions. Diversion of imperial resources changes development of British territories in India and SE Asia, which changes the British calculus in Africa later. Imperial emancipation of slaves may be affected, but it's likely that it continues apace with interesting consequences for the agriculture sector in the western Mississippi Basin.