The nicest place in the centre of town (Zone 1 of the Underground) is Bevis Marks -- The Restaurant, which is actually inside the sukkah/adjoining conservatory of Bevis Marks synagogue (oldest synagogue in the UK -- built in 1701).
The nearest tube station is Liverpool Street.
Not sure if it's glatt, but since the hashgacha is Sephardi Kashrut Authority and Sephardim eat glatt I'd imagine that it is. Why not email and check?
[email protected]
And the website is:
http://www.bevismarkstherestaurant.com/
It's expensive, but really very well worth it.