As I said, my GP is great and genuinely concerned about my skin considering my history. But a routine skin check is not done on the NHS. As BD said, if you have a specific mole that you concerned with, you'll be referred, etc. I've had two melanomas removed and neither looked odd to me (and I suspect a GP would have also dismissed them). And I like them to check everywhere, particularly my head (under my hair) because I would never see a mole there!
I did ask my GP what happens if I pay privately to go to a dermatologist and he/she discovers melanoma that requires full blown cancer treatment. He said I would then be referred back to the NHS for treatment (so I wouldn't have to pay for a lengthy cancer battle). I would still be responsible for all appointments, lab tests, treatment prior to the NHS referral though.
Hopefully all of this is hypothetical and I "pass" tomorrow's £250 skin check.