London Bridge tube station had to be rebuilt as well (not too add a mezannine per se but just to have enough space to hold the passengers waiting for trains), in the late 90s to house Jubilee Line but also to expand the Northern Line platforms. Even this 20 years later looks too small. Both these lines operate every 2 minutes (Jubilee slightly more than every 2 mins in peak).

To add to what the other commenter said - Bank station (the busiest on the London tube network) is getting rebuilt too, to add a "mezzanine" type area for the Northern line. Through-trains, all to the same destinations, are every 2 minutes and the platform is frequently closed due to overcrowding. To the point, in fact, where there's often a one-way system enforced in the station, or people are kept at street level.

Ultimately it's usage levels and capacity. UK stations don't have the capacity for the passengers, and often the capacity on the trains either.