This one is based on an outline from a post on Ken Hite's (GURPS alternate worlds contributor) site
http://princeofcairo.livejournal.com/
This is a world where Burr does end up taking over Mexico (to quote Hite, "
British gold, Spanish fecklessness, criollo opportunism, if you're wondering how it might have worked") and ends up becoming Emperor out of frustration with the effort of making democracy work in Mexico.
Flash forward a century and some change, and we have a world where the US broke up as a result, and the three biggest powers are Napoleonic France, the Russian Empire, and a Britain which managed to stave off Napoleonic invasion but was pushed off the continent, and has made up for that by expanding its empire even further than OTL and making a concerted effort to develop heavy industry (for military muscle) in its colonies far beyond anything OTL. Mexico (which has seen political stablity, if of a somewhat repressive sort) and far more emigration than OTL, is, along with Japan, an important second-rank power. The US, poorer, more militarized (it has _three_ unfriendly borders) than OTL, and politically messed up to boot, is largely isolated internationally, and barely counts as "second level".
There have been wars and military clashes, but no full-blown apocalyptic multi-nation wars on the scale of OTL 1914-1918, but this may change. Russian industrial growth is threatening to upset the balance of power. Meanwhile, the Ottoman Empire, where the Sultan is now a powerless figurehead for a Committee-for-Union-and-Progress-like military dictatorship, is suffering internal unrest: and neither the French, the Russians nor the British wants the others to hog all that oil...
Bruce