i have skin cancer on both sides of the family, and I admit to burning a few times as a child. I keep a careful watch on all moles, and a few years ago one started to look a bit funky. I went to my GP, who said that although she was pretty sure it was fine, she would be happy to refer me to a dermatologist.
I could have seen one through my private insurance in 2 weeks. Instead I waited 5 for a NHS appointment (which was with the same Dr, just at a different hospital!). I was happy to wait b/c I wasn't overy concerned.
When I went to the dermatolgist, he gave me an 'open referral'. He said that because of my fair skin, family history, previous burns, lots of freckles/moles, I was more at risk. The 'open referral' allowed me to make appointments directly with him for 18 months. I never used it, and would now need to go too the GP if I had any other issues.
My advice is to go to your GP early on, bring your previous notes, explain that you'd like a check every 12 months, and see what they say. Also, please know that since you've had prior skin issues, any future skin issues may not be covered by any private insurance you take out in the UK.