We had to act right away on the peeling back wall, so installed a not-very-expensive wraparound tub wall. It will remain for now, though at some future date I'd like to replace it.
Two years later, after attending to more important alterations, we carried out the old vanity (a difficult job as it was heavy, constructed of very solid plywood), and stripped the plastic tiles from the drywall behind (an easy job as they practically fell off at a touch.)
Michael detached the toilet, which we moved temporarily to the back porch, then removed the linoleum tiles on the floor, the plywood beneath them and finally the fir boards below that to get down to the subfloor.
He replaced the plywood, laid heating cable over it,
poured self-levelling cement (the worst part of the job as it didn't self-level willingly), set the new terracotta tiles and finished with a dark-coloured grout. I put two coats of sealer on the grout.
Then I repainted the walls, we installed beadboard wainscotting, which I had already painted out in the backyard, and replaced the toilet. Our new vanity including sink and faucet was acquired from Craigslist, and suited both our colour scheme and the rustic look we were aiming for.
Our old oval mirror that we've had for many years turned out to be a good match for the vanity. We added a new curved shower rod and a half-curtain to the window in colours that matched the walls.
An inexpensive print on the wall and a bamboo etagere to hold towels completed the room.
Cooler weather in the last week gave us an opportunity to test the floor heat and find it is working well. All in all, we're pleased with the transformation.