It is heavy work, but the mudbed and tile will cleave off of the bottom layer. A demolition hammer - Hilti makes one, can be rented for a day and will chop it up into manageable pieces. That and a couple of wreching (crow) bars and it should come out. Watch out for the metal lath, it is quite sharp. If there are areas with only wood products in layers you want to remove, the easiest way is to use a circular saw (use an old construction blade that can handle nails), set to the depth of the layers you want to remove, then saw it up into reasonably sized pieces, then pry them out. It gets a little messier at the edges of the room if the sheets run under the partitian walls, but they probably don't, only the planks probably run under the walls.
Over the top of the planks, first, make sure that they are well attached to the joists. If any seem loose, squeak, or obviously move, reanchor them to the joists with deck screws. Then, install, preferably, 1/2" plywood over the whole thing. At that point, you could build it up further for tile, or vinal. Normally with vinal, you'd put down a layer of luan ply, fill the joints and nail holes if required, and then put down the vinal.