I strongly disagree with those thinking there would be less encroachment into Indian lands and Spanish America. A British America is going to require most power being relinquished to local power, as the empire did in India and Africa. The reason the Indians were treated better in Canada was because it was in the economic interests of the settlers to work with them in the fur trade. The Brits were perfectly happy to push natives off their land, as in Kenya, or a spot of genocide, as in Tazmania, when it suited them. The same will happen with the Five Civilised Tribes in our timeline.
As for the push from London, it will be around merchant commercial concerns, and the focus there was picking apart the Spanish Empire in the 1700s rather than in the 1800s as the US did in our timeline. The Caribbean will be the main focus, particularly Hispaniola and Cuba, but places like New Orleans and Veracruz would also be of interest. I imagine in this timeline the settlement push would be to the south west rather than the north west. That probably results in Mexico being cut off from its northern territories sooner, and at a more southerly point, but later settlement of the inland great plains and rockies. California will be settled from the sea but Russian territory might be left as is, as war with Russia will be seen as more trouble than its worth.