Well, if the ARW results in a British victory (for whatever reason), and the Americans throw a second revolution (see General Finley's timeline, The Federal Republic of America) which succeeds, then the French Revolution is delayed and Napoleon butterflied. As such, the Spanish still have their pretty solid claim to the whole of Louisiana and their sphere in the Americas.
When the French Revolution eventually does roll around (that's an 'if', by the way), something similar to the Continental System or a continental blocking of British trade might occur,forcing Britain to look for other markets. So, as in OTL, they support revolutionary ideas in Spanish America and begin a campaign of shit-stirring.
When Mexico eventually revolts agains Spanish rule, the British invade Lousiana and take over Northern California in the fray. At the end of the whole mess in Europe and the Americas, Britain hasn't budged and they annex their occupied territories while being the first to recognise the independence won by those who fought for it.
So not only has Britain managed to annex Northern California, they've also stunted American westward expansion and have very good relations with Mexico and the Caribbean.