Even for private health care, you usually see a GP first (could be a private GP or NHS, in-person or video) and they then refer you to the next thing whatever that may be. ( N.B. that a NHS GP can give you a referral that you use for private treatment)

There are some things you can just straight up book an appointment for yourself without a referral from a GP, but 95% of the time you start with a GP.

No idea specifically about dermatologists, but my expectation would be that would be the sort of thing that would need a referral for. Perhaps for some "non-medical" procedures and 100% for cosmetic procedures you don't need a referral, but anything even tangentially close to The C Word would almost certainly be sending you down the normal channels.