For football, not sure if this is possible. England vs Scotland was one of the first international matches, and the English FA is the first FA (hence just being called "The FA").
For rugby however, I could have the Lions becoming the primary international side. However, an international side will be dominated by the English due to population size being vastly weighted in their advantage. England, Wales & Scotland (and maybe Ireland depending on whether they participate fully in a proper Lions team) would just remain for Home Nations tournaments now and then. How exactly that happens I'm not sure; the 6 Nations is one of the premier rugby tournaments that has grown out of a Home Nations tournament - I suppose the tournament would need retarding to allow the Lions to take premier stage.
For cricket, it's technically "England and Wales" already, so we're half way there. Could possibly be expanded to England, Wales & Scotland due to the size of the sport, and renamed the British Cricket team.