HeavyWeaponsGuy
Banned
OK, how about from the very beginning of Mexico's existence as an independent nation the UK becomes very friendly with México City, no matter if it was a Republic or an Empire (the only concern here is ensuring stability). In effect, Mexico ends up becoming an "honorary Dominion", with the effect that Mexican-*Canadian ties end up becoming stronger as a result and both British and *Canadian businesses thrive in Mexico alongside local businesses (as well as the case of people later on who have impeccably [admittedly old-fashioned] British names and surnames who don't speak a word of English at all, or ditto with impeccably French-Canadian names, just Spanish). As a result, both British culture and Canadian culture (English-Canadian and French-Canadian) rub off onto Mexicans to the point where they are considered integral parts of Mexican culture.
Even in that case, I still wouldn't see Britain snatching up Alta California - at this point, unless if one was Canadian (NWT, anyone?), Britain was basically done with the whole North American land thing. Even if there were *Canadian hordes descending onto Alta California and Nuevo México. If anything, Britain might end up defending the Mexican claims over the American claims.
Well, given Britain's other colonial adventures in the area (mostly Oregon) it quickly became clear that aside from Canada, Britain's possessions or dominions in the region were going to be swallowed up by the Americans eventually, if not by being outfought, than by being outbred (American MO for getting a territory was move in settlers and eventually have them able to overwhelm whoever the original owner of the territory was, they did it in Texas and pretty much all the Mexican Cession territories.
I can see a more realistic scenario where Britain extensively supports Mexico and even sends military aid and advisors to give the Mexicans more up-to-date equipment (their army was pretty decent though their equipment cost them dearly in battles) but doesn't actually intervene. If Anglo-American tensions ever get too hot the British know they have a protracted overseas conflict in which they stand a decent chance of losing a lot of their American possessions, it isn't so much an issue of is Britain a better military power than America it's an issue of can they handle the logistics of a land war against the USA and are they going to make the commitment in both manpower and money to win. Individually, they are quite difficult problems, together they make for a very unappealing situation for the British if they have to go to war.
In the longer-term they would have to consider the implications as well, another war with America so soon after the War of 1812 is only likely to make America MORE contentious with Britain and worsen existing tensions, Britain wouldn't want to be conducting expensive wars with America on a regular basis.