Tabby - hard to give you an answer w/o knowing how long you plan to be back in the UK and what your future plans are with your girlfriend.
Right now there are essentially no entry-level jobs for most people. There's a massive over production of instructors since 2023 and backlogged employment means incumbents are not leaving jobs. It's taking many months to find an initial CFI job. And anecdotaly student starts are down.
There are no "work visas for pilots." There are no "I've got an American girlfriend visas." If you are married you can become eligible for a green card. Most worthwhile American pilot jobs "require" a degree even if a degree is "not required." Your competition will have one...
If you complete a frozen ATPL program in the UK and fly enough to accrue 1,500 hours you *could* possibly get an airline job in the US once you have work eligibility. You'd be more competitive than "1,500 hours in a 172." Are you going to remain in the UK long enough to do that? This is your ace.
If you get the right to work and an FAA ATP you're pretty competitive. But be aware you won't be eligible yet for anything that requires X number of hours of Part 121 time since "I flew a jet in Europe" is not under the FAA's Part 121 of the Regulations.
Properly planned conversion from UK ATPL to FAA ATP is neither hard nor time consuming. Certainly easier than the other way around.
The only place you have the right to work is the UK. The only place you should train is in the UK. If you are likely to get married, or once you do, if you have an airline job consider working on an online US degree.
You have a lot of "if" pieces in your future. Make a plan. Get smart on the process. Marry for love. Don't borrow lots of money. At least get Private done before going to any "airline oriented flight training." Good luck.