No one knows which side of the stairs to walk on. Not ever. Not even with the friendly KEEP LEFT signs at the top and bottom of each flight of stairs in the Tube. Not while entering and exiting buildings either. No one, ever.
It must take a particular kind of fortitude to deal with the influx of confused foreigners, if you are a person who lives in London. All you want to do is go into a museum. All you want to do is walk up a flight of stairs from your subway ride without dealing with an obstacle course of would-be shoulder-checks.
No one knows. I never figured it out. I tried to do as the signs said, as the traffic would indicate, but all I ever did was dodge, a leaf on the wind.