Suppose Great Britain has New Orleans then!
And it's 1806, and they won't leave! The US probably cannot dislodge them. Yet! There would have been a very definate anti-British bias in the US Government for quite some time. Eventually, American population pressures will force a confrontation. The Southern slaveholders , having been deprived of Louisiana, Texas, Arkansas, and Missouri are going to be all for getting the British OUT! Northern Developement into the Ohio valley and into Illinois and Wisconsin will place pretty good sized US populations right there in mid continent, right where it would be hardest for the British to support any claims! After all, The British may have New Orleans and the MOUTH of the Mississsippi, but they don't hold it's eastern bank, so it's going to be very tough using the Mississsipi River as an avenue to project force! I'd say that by the 1860's, instead of a Civil WAr, there would have been a war with Great Britain for control of all, non-hispanic North AMerica, to include Canada! A war that by 1860, the US would win!