Getting the team to be competitive wouldn't be too much of a problem. You could've made many England teams highly competitive by adding some of the other Home Nations players - you can make any England team from the Venables era to around 2005 much more competitive simply by adding Ryan Giggs on the left wing, and if you make Alex Ferguson manager then you've pretty much guaranteed a major tournament win. Even now the England team would be better by adding Gareth Bale and Jonny Evans.
Getting the political will for this would be more tricky, and either needs a pre-1900 POD where the Home Nations aren't established as separate or, as has been mentioned, England not to win in 66 and the rest of the home nations to go into similar decline in the 1970s (particularly Scotland, who were the only representative in 74 and 78). That, along with pressure from FIFA, might allow a unified team.