on a tangential note, according to wiki (so take with a few tonnes of salt) shortly before the Mexican-American war the Minister in Mexico wrote to Palmerston urging him to establish a colony there, so if its true there's another possible POD.
anyway, I'd say that at this point Brittan would totally dominate the Americans in any war. probably the usual "blockade the east coast, ruin economy" plan, with a peace that just recognizes British dominance in California. Also the Monroe Doctrine takes a bit of a hit.
If the U.S. goes for a longer war, then it will probably end up as either some compromise peace of exhaustion when something more important to the Brits comes up, or possible an American cessation of the Oregon territory or part thereof. In that case American access to the Pacific could be gone, which has some interesting effects..