Opus One (in the Radisson Edwardian) is popular but more in a nice bar, similarly places like Cloud 23 (on the 23rd floor of the Hilton but you usually have to book).
There are a number of tea rooms I'd go for over somewhere like Harvey Nics.
Richmond Tea Rooms - lovely and Alice in Wonderland themed:
http://www.richmondtearooms.com/
Maybe more a girly thing but I love it, went with friends for an afternoon cream tea treat my last birthday. I can verify that the scones are absolutely gorgeous!
47 King St is a bit more posh:
http://www.47kingstreetwest.co.uk/
Teacup on Thomas St is very popular but service can be quite slow and it does get busy (often a queue to be seated). I go there fairly often but it wouldn't be my top choice for a special treat afternoon tea.
http://teacupandcakes.com/
Pattiserie Valerie does amazing cakes but not quite so authentic. One of those places that makes you drool just by walking past the window!
http://www.patisserie-valerie.co.uk/