I've actually researched this myself. Yes, it's possible, but you'll need to go through a semi-lengthy licensure evaluation post PhD. You'll also, most likely, be looking at NHS jobs. I believe, and I still don't have a definitive answer on this, you'd start at the principle psychology level with your PhD and additional training (you can Google the pay).
It seems like a solid lifestyle, decent pay if you're not in London, and could offer a fun change of pace. The biggest sacrifice seems to be the reduction in income, but I 100% could see the appeal if you don't plan on having kids.
Scotland and, really anywhere that isn't London, has a housing market a clinical psychologist can survive in (you can get places that are pretty nice). Another huge pro is your vacations around Europe just got exceptionally cheaper.
The biggest piece of advice I could offer is finish your PhD and US licensure first. Even if you never intend on coming back this is not something you'd want to go without, a solid safety net and it may make you eligible for other European gigs (the Netherlands loves their US Clinical PhDs to fill their research positions).
Edit: This plan is contingent on keeping debt down (or having none at all)! Be mindful of this while looking into programs.