I think you may be in luck.
With a 1925 home, you could have lath and plaster instead of drywall (gyproc). Looks like you have drywall. IMO, what you have is wallpaper that was put up with a water soluable glue, that was painted over. When the moisture problem started, the wallpaper is delaminating, and leaving the drywall behind. Because you can see the grey (and the nail head in the blue room picture), this means that removing the wallpaper should not be a nightmare for you.
I would start with the worst room, and score the wallpaper (they sell tools for this) to get through the paint / possibly vinyl surface. Then steam the room / apply damp rags to the walls to moisten the glue, and 2 hours later, remove the wall paper. When it works, the paper just peels away without scraping....
For mold issues -- you need to cut away and dispose of moldy drywall, paper, carpet (if any). But the wood can just be treated with a bleach solution (or mildew solution), left to dry, and recovered or painted with stain block. The mold is on the surface of the wood, and as long as you don't have rot, the stains may be visible, but you kill the mold.
If you have a fair amount of mold, I would wear a mask while tackling it, as it can cause you to cough for a few months after due to quasi-allergic type reactions even if you are not sensitive to mold already. That said, I only started to wear a mask for this type of work, and I am fine.
Images -- easy wallpaper removal of vinyl paper with water soluable glue; wallpaper removal tool; ugly old wallpaper that requires excessive scraping