There all sorts of London cool, however linking simple food and cool is quite tough really.

A few ideas:

Sketch - vibrant design, celeb crowd, mediocre food
The Ivy - simple comfort food, lots of celebrity type people (though how many US citizens would know them?) but would disappoint with design
Wolesley - good food, resembles a Middle Europe grand cafe, great atmosphere
Hakkasan - decent rendition of Chinese food, very cool feel

I doubt conventional afternoon tea would tick the boxes (it can feel a bit stuffy), but I think Sketch do a modern version.