If you happen to have a dental emergency, I recommend you visit the acute dental care/emergency clinic at Guy's Hospital.
It is located on the 21st floor of the tower. I advise you go there early (08.30) as it can take some time to be seen by the triage person.
Try Borough Dental Practice, 305 Borough High Street. It's NHS but does private work too. I had two front teeth crowned (replacing NHS ones) and some old filings replaced with white ones on a private basis two years ago and was so impressed by the care. The cost was so much more affordable than quoted privately elsewhere and just to give an example of the care I received, I was given an appointment at the place where they made my crowns so I could be properly colour matched in a daylight environment, I learnt that this so important and in my case was overlooked in the past. I was fitted with excellent crowns which nobody notices as being 'fake'.
The place looks a bit glum and basic from the outside but don't let that put you off!
My experience is based on my treatment two years ago and I trust I would still rate them as good if I visited them tomorrow. I've popped a link below to their NHS Choices overview, there's some reviews there.
I agree all comments re Borough Dental Care, corner of BHS and Trinity Street. Nice, friendly receptionists, good dentists - actually clean/polish your teeth as part of check up and told me I didn't need to see the hygienist unless I wanted to (no rip-off there, unlike Talbot nearer London Bridge) Open early and on Saturdays. What's not to like?