I need them for the walls of a gym room that I've built.

Tried Thai Watsadu, HomePro and tile shops, but none seem to stock plain mirror tiles (maybe 30cm or 40cm square size).

Any suggestions as to where I might buy them?

The only mirror tiles I have ever seen anywhere were made by Pilkington but now do not seem to make them. Try Chalong Glass & they will make mirrors to any size you want.