Across the Board: Around the World in 18 Games is an exhibit sponsored by the British Museum that is currently touring Great Britain. Being shown are artifacts of such ancient games as Senet, Nine Man’s Morris, and Ur, as well as the Lewis Chessmen. If you click on the Games link, you can actually play the first three online as well.
It doesn’t sound as extensive as the Asian Games exhibit that was in Washington, D.C. last spring (missed it, alas), but it might be worth the trip if you happen to be close by one of the stops. And it sounds like a lot of the artifacts are from the main collection of the British Museum, so I suspect that if you were to go there after the tour is over you could see them there, albeit spread through the various halls.
[Edited for clarity]