It's not in SE1, but not too far away... Wandsworth Dental Centre on Wandsworth Road SW8, which is about 5 mins from Vauxhall on the 77 bus. They do take new NHS patients, and the dentist I see is very friendly. I don't have any connection to the surgery other than being a satisfied patient.
There is also a new NHS dentist about to open on Horseferry Road, SW1, close to Lambeth Bridge, on the C10 bus route.
Wandsworth Dental Centre will definitely take paying NHS patients. I joined a couple of years ago, and several colleagues have since joined. They also don't have a catchment area. And their hygienist is £35, too - although I didn't realise this was cheap when paying for a couple of sessions!