What to do with wall behind bath taps
Don't even worry about asbestos, it's only a concern if you tear it up into a breathable dust.
You say you are concerned about water intrusion, but all around that area there are gaps and cracks through which water could penetrate. You say you are worried about mold, but the wall above is covered with what looks like bio growth (mildew).
I recommend that you:

clean the area with the missing tiles with TSP, then a bleach solution (in fact, do that the entire area including the wall).
seal all cracks, crevices, and edges with paintable caulk
paint area with missing tiles, and the plaster wall area above, with quality indoor glossy bathroom paint.

If you had more experience with intrusive DIY projects there are a lot of other, more permanent solutions (e.g. redo all tile) but at your skill level this should help and it will look good too.