I'm not sold that any proposed UK control of California, or even taking an independent California as a protectorate, in the 1830s/40s leads to the UK and the US carving up Mexico from Tampico or Guaymas northward with all of Oregon Country going to the UK.
As I said in a similar discussion on a larger Mexican Cession, there are myriad domestic issues in the US at this time that would only be worsened with the acquisition of more territory that could in theory (and probably intended to) be turned over to slave power. Now, imagine a larger Mexican cession but coupled with territory that might be more amenable to Northern, free labour settlers being given over to perfidious Albion. Think even in the unlikely event this is what happens the most interesting thing to come from it might be driving towards a Northern secession.
Then there's Mexico, which will not be a passive victim of history at the whims of the UK and US ITTL any more than it was OTL. It won't be able to fight both Washington and London at once, but I don't see any scenario in which that situation occurs. It's dependent upon both the UK and the US being on the same page and a casus belli which will draw both of them into a war at the same time. Perhaps more likely is that any shooting war that develops between Mexico and either the UK or US results in the other offering up support to Mexico in exchange for favourable borders.
The situation in the territories under contention too was by no means monolithic. California has the Californios (with those in the north and those in the south having different priorities); other Mexicans who might be more inclined to Mexico itself than any combination of the UK, US or independent California; American settlers split between Northerners, Southerners and Mormons. Texas, still theoretically independent at this stage, was in negotiations with Mexico and UK after annexation efforts in the 1830s stalled. And Oregon Country is dotted with British and American forts and settlements all over the shop.
Can actually see the opposite situation, that disputes over California/Texas/Oregon lead into an all out multi-sided war in the 1840s, happening instead of UK/US divvying up the west. That itself is an interesting prospect from an alternate history perspective maybe worth its own dedicated discussion. Outcome from that to a "British" California might be any peace settlement where the independence of California is guaranteed under British "protection". The US would likely still get Texas in that outcome but it might be a far smaller Texas than what the US claimed OTL. I can see Mexico retaining more of their pre-war territory north of the OTL Guadalupe Hidalgo border. Oregon would still be split, though who gets the lion's share depends on the situation when the war ends - in that aspect could see the US being granted part of Alta California north of the 40th or 41st parallels in exchange for not pressing territorial claims on Vancouver island or even north of the Columbia River perhaps.
A lot more you could go into in such a conflict, such as conflicts that arise from it within the US (over slavery), British North America (the 1837 Rebellions and the Caroline affair are very recent events), and Mexico (conservatives vs. liberals). Could also speculate on the prospect of France and/or Spain getting involved (the Pastry War is another recent occurrence, and there were Cuban independence efforts in the 1820s and 30s even before US filibusters got involved).