Like most ward sisters, tired or otherwise, you clearly know very little about how general practice operates.
The 2 week referral pathway works in exactly the same way for dermatology as for any other speciality. The whole point of 2 week referral pathway is that certain, high-risk symptoms are prioritised. No speciality accepts lower risk referrals via 2 week pathways. That is true of every single 2 week pathway, not just dermatology.
You're also ill-informed about doctors' training. All doctors undergo extensive training on the skin as an organ, starting in year one of medical school, with the anatomy, physiology and pathology of the skin. By the time they qualify, many months of training, cumulatively, will have been spent on issues relating to the skin - not just purely dermatological conditions, but the many other pathologies that can have skin-based manifestations.
GPs undergo further training as part of their GP rotations. Some of them actually spend months working for a dermatologist, and all of them spend a lot of time learning about skin disease in general practice.
After qualifying as GPs after a minimum of 10 years training, many of them go on to do additional post-graduate qualifications in dermatology. As well as running specialist clinics in general practice, many of them are employed by hospitals to work in outpatient clinics too.
The evidence is that GPs are as accurate as diagnosing skin cancers as dermatologists - this is a link to an Australian study (many GPs in Australia are UK-trained), and the findings have been reproduced in the UK and other countries.