Interesting I was walking near mansion house (in the City) yesterday and could feel something rumbling underneath my feet through the pavement which I assume was Crossrail.
Some of the older lines such as the Met and District are subsurface rather than deep. I believe some roads can't take truck based cranes because the tube is only 5m below.
Most likely that was the circle/district line. It's only one flight of stairs down from Cannon Street and Mansion House. Not sure how many metres, but it isn't far down at all.