I contacted the Museum of London and received this helpful reply today:

"You enquired about the London Stone. The building that holds the stone is due for redevelopment. While this is happening the stone will be displayed temporarily at the Museum of London. It will then return to an enhanced display in its original location. I have no exact timescale for this process at present. So rest assured that it is being well looked after and its situation will improve.

Hedley Swain
Head of Early London History and Collections
Museum of London"