if we update the playroom we'd just replace the fiberboard with drywall. if we do the bathroom it would be updating the old subway tile to match the new (behind the shower) and replace the floors. fixtures would likely remain.
I say bathroom. It looks like it needs it more. I know you want to keep fixtures, but I'd ditch the pedestal sink and go for a small vanity. There are Euro-style ones with small cabinets under the sink.
Since your son is a child, you may also want to update to a single lever control that has scald protect. That would bring it up to code and ensure that the water wouldn't get hot enough to burn him.
Delta makes a handle with two levers stacked on each other (Lahara dualstatic) that lets you control pressure and temperature separately. Requires a Delta Multichoice valve. They come in threaded, ones needing copper sweated, and pex barb. My friend has one and it's good. I'm putting 2 of them in my house.
The bathroom for sure. Not only does it look like it needs a good renovation, the layout looks weird. Is there a way to get the plumbing all where it is supposed to be? Normally the tub/shower valves are on the end wall where the shower is. The toilet and the sink are supposed to be on the same wall. (it looks like the toilet is on the wrong wall) I don't know what happened here. I see handicapped bars, but it would be difficult to get a wheelchair through that maze.
Bob, I agree.
Btw, I'd keep the grab bars for later use. Grab bars can be expensive.
Since the bathroom is for kids, I would also consider having a handheld showerhead that they could reach. Probably not in the budget but KES Rainshower has a rainshower head with a built-in slide bar and a handheld showerhead that can go up and down the slidebar. The switch from handheld to rainshower is at the bottom of the slidebar. My friend has one and it's very nice. The one I linked is about $70. They are available in different finishes though.