I'd sooner expect a real difference to affect Mexico.
As in, with yet another US-British war, the Brits will be more likely to back Mexico against the USA (even if only for a decade), which is bound to make US expansion into Mexico more complicated. Of course the great push of colonization is still westward, and Mexico is still busy failing to control anything outside the Mexico heartland, so it's not the same as saying Mexico keeps California etc. But you might see more multi-step processes and maybe a Californian republic separate from both.
As in, with yet another US-British war, the Brits will be more likely to back Mexico against the USA (even if only for a decade), which is bound to make US expansion into Mexico more complicated. Of course the great push of colonization is still westward, and Mexico is still busy failing to control anything outside the Mexico heartland, so it's not the same as saying Mexico keeps California etc. But you might see more multi-step processes and maybe a Californian republic separate from both.