A few years ago it seemed it was going to be moved. The London Stone is on Twitter (!) and appears to be saying WHSmith have taken over the shop behind it. "London Stone @thelondonstone 15 Nov "I am not sure i like my rear end being obscured by a magazine rack these days..."
https://twitter.com/thelondonstone
Bumping Andy's comment further down this page :-
More discussions on the London Stone in our Forum
http://www.megalithic.co.uk/modules.php?op=modload&name=Forum&file=viewtopic&topic=3844&forum=1
Something is not right. This message is just to keep things from messing up down the road