This is very much a work in process! Please leave a comment below if you have a bookswap shelf/stand/etc to add, and feel free to copy the idea for your own region!
Natura Café, Goldsmiths' Students' Union (first floor), Dixon Road, New Cross, SE14 6NW
The Camel and Artichoke (upstairs), 121 Lower Marsh SE1 7AE (near Waterloo)
Books for London (bookswap shelves at major stations)
Books on the Underground (book drops on the Tube)
Little Free Library Project UK (essentially, bookswap shelves in mini houses in people's front gardens)
And don't neglect your local library!