IMHO, GB might promote secessionists ambitions of New England, and create an useful ally in the area. They might be willing to annex Louisiana, and New Orleans, though. Or again to create a new republic there. The western border of USA would not be at the Rockies. I wonder also what will happen with Florida here.
When Texas time comes, there are two obvious plots for GB (in particular for a GB which is still considering herself the Mistress of north America): support an independent Texas (might be tricky, but it might be done. And promote immigration from Europe, to create a nation with no tie to the USA) or support Mexico, which is easier (and the price might be paid just a bit later on, when California gold is discovered). In both cases, there is even a moral justification: keep slavery out of Texas. This would be likely to be a very different world, with GB much more present (and dominant) in NA, and in the Pacific too (there's no doubt that Hawai'i would be annexed very soon in TTL)