I always saw an OB/GYN in the US for the annual pap smear & breast exam - just seemed to be the way things were done there & it never occurred to me to question it.
I've never been to one of those FPCs here, but I haven't had need for contraception either. At my local surgery, it's the practice nurse who does the smear test - not a GP. I've been fine with that, although at first I didn't really understand why the nurse is called a 'Sister' - because she's not a nun.
Apparently, in a couple of years, I'll also receive an invitation to my first mammogram on the NHS. I had a baseline mammogram in the US when I was 35, and then one each year thereafter until I moved here, so I was a little surprised that (all things being normal) you don't have them until age 50 here (unless you have a reason for needing them sooner).
I was on a blood pressure tablet when I moved here, and the GP here just looked up what it was & prescribed me the same thing (by a different name) -- I don't take that any longer as my blood pressure came down through lifestyle changes. Also over the years, I have taken a couple of different anti-depressants that were known by different names here compared to the US. One time I asked the GP about taking Wellbutrin instead as it appeared to have some appetite suppressant properties (vs the other one I was taking at the time having had the opposite effect!) which it was being used for in the US - however, it was only prescribed here for smoking cessation & nothing else. So there are things a person might run into like that.
My experiences here have been mostly positive, and I agree with DrSuperL99's advice to just wait & see & experience it for yourself - trying to keep an open mind, and maybe it'll be better than you think.
I did spend a night at A&E once here with a knee injury & it wasn't very nice/clean there (it literally stank, the exam room wasn't clean & they were allowing an abusive drunken person to wander around far too long before security finally ran him off) -- which makes me NOT want to ever go into hospital! But I just have a thing/fear about hospitals generally as well. We do have a little private health care policy through DH's work, so *fingers crossed* if I ever needed the hospital, maybe I'd get into Nuffield -- people tend to come out of LGI (if they come out at all!) sicker than when they went in. But you could say that about the hospital in the US where my mom used to live too.