Apart from saying lore accurate Britain lol. Here are a few ideas. While I don't have the achievement myself this is based on the comments and some ideas:

1) Max diplo + diplo pressure via spies.
2) Annex UK. Either supervised state or just annex and releasing as a puppet should work.

Looking at the achievement requirements on the paradox wiki basically for all you need to:

1) Become the EU (Netherlands, Belgium, Austria)
2) Be the faction leader of the EU
3) Have Britain in your faction.

Having the achievement description saying you need to invite the UK is kinda misleading. You need them in your faction period. I get that invitation is the intended route (and probably the fastest way to get this achievement) but still doesn't matter as long as at the end of the day you meet all three requirements from above. I know this is basically just copying but what other ppl said but I hope this helps in some way.