with the tile on it, and are successful in not beaking the tiles, you can soak the sections of underlayment and tile in just plain old water. This will take quite a while, but eventually the wood and the grout will soften enough that they let go of the ceramic tile. You'll still have to clean the backs up somewhat with a good stiff scrub brush, but the tile should be in pretty good shape and can be relaid. If the tile was originally attached to a mesh backing, the backing will also soak loose, so if this was a mosaic, it will be more tedious to put it all back down.
I have done this using a child's plastic wading pool. Just through those pieces of those tile-faced underlayment in the pool and keep filled with water (I even added a bit of detergent just to discourage varmints!). This may take several weeks, but it sounds like you have many, many other things to do for this remodel anyway, so they should be ready for you when you're ready for them! Good luck.