If Britain keeps the U.S., do the Louisiana Territory, Texas, New Mexico, and California eventually

    If Britain keeps the US colonies then I think its almost certain to take over Louisiana simply because between Britain's military strength and the population of the colonies such a 'empty' land is going to be taken over at some time in one way or another.  Either by seizure and annexation in a war with France - or their Bourbon ally Spain after 1763 - or by a flow of settlers from the British colonies who effectively do a Texas, but in far greater numbers.  Plus given the importance of the Mississippi to the development of the territories of eastern Louisiana which were gained in 1763.

  For the other lands they may stay in an independent Mexico depending on a)  Its relations with Britain and b) how tightly Britain is able to control its own colony to prevent them again swamping fairly empty lands bordering the empire.  This was a problem in places like Africa and Australia at times as often local white settlers would clash with local people over resources they wanted and London would have to step in.  This is also why, although I hope it ends better for the Indians that the proclamation that secures Indian lands west of the line won't do more than delay and hopefully control the degree of western expansion by colonists to minimise excesses.  It might end up with a war which sees much of northern Mexico annexed as OTL or with more peaceful relations and relatively stable borders.

  The exception might be California in that when gold is discovered your likely to have a flood of European settlers, many of which will be from the UK/N America and that will lead to questions of the government of the region.  Unless TTL Mexico's has a much more stable and efficient government and has successfully settled the region itself more securely I can see that also becoming part of the empire or possibly an independent state.