The Cadogan Clinic on Sloane Street does mole removal. I haven't been there for this procedure so can't comment specifically but I have seen John Ferguson for another skin matter and would recommend him.
I have found Dr Harland at Sk:n clinic at Epsom Hospital very good for cosmetic mole and skin tag removal. It's a bit of a trek, but worth it. I originally saw him at parkside but he does his cosmetic work from Epsom. https://beta.sknclinics.co.uk/clinics/t ... owan-house