I can't say I ever thought that having your nails buffed up was particular important when meeting someone, but then again, I'm not into that sort of pampering. There are loads of places to get your nails done, and while I couldn't tell you which is best, because I don't use them, I would say that you could probably get just as good a job from somewhere less expensive as somewhere high priced, so best not to go searching in the really expensive parts of London.