I flew commercially in the UK under EASA / UK CAA and transferred to US FAA. Like the posts above, it's a combination of aviation and life advice. It's great to be in a relationship where you are making plans and looking to the future, but as with being a pilot - don't hinge everything on that and have contingency planning! What does that mean? Have a plan that works whether the relationship lasts (or not). Maybe don't phrase it to your partner in that way!
Given you are in the UK and don't have a green card yet, I would recommend getting established with the UK route. The ATPL exams are not fun but they're not entirely worthless and you'll learn something, and if you can set aside time for them they're not too bad. With a UK airline job, you'll build experience in a way US pilots can't do (they usually need 1500 hours to fly for an airline and you don't). In a few years you'd be able to do the UK CPL / ATPL conversion to FAA which is quite painless - and that opens up the opportunities in the US which are vastly greater than the UK.
That also works with your life plans as there is time for your relationship to develop and possibly marry - then you are positioned for the green card, without forcing the situation to meet the licensing and employment route - which would not be a good idea, I would suggest.
Good luck with it, exciting journey.