As a complete guess I'd say you'd be looking at a few hundred minimum.

We have an NHS for a reason - go see your Doctor.

I'm actually looking for the name of the clinic.

I know about the NHS but the referral times are just too long.

I live in leeds and went to my gp and was referred straight away while i was in the room.They actually asked me which clinic i wanted to choose and i was given a printed letter and told to leave it 24 hours then phone clinic to book appointment with dermatologist.Are procedures different for being referred in london then?
In order to be referred though you have to have tried all creams and antibiotics first.
Im sorry i cannot help with clinic names and prices in your area ,maybe if you can just google london dermo clinics and look on their websites for a price list,but either way i would have thought it will be pricey.
good luck anyway