How to deal with a failed primer coat
When you say redo the entire bathroom does that just mean redo the sections of exposed wall? Because if it doesn't you should consider the opportunity to pull the old drywall entirely, replace it with a modern bathroom wallboard (such as densarmor or similar) and refinish the walls. Sanding and reskimming are messy and laborious, replacing the wallboardb would give you a better result and maybe afford more opportunities along the way.
But let's say you're not going to replace the wallboard.
You should sand them down aggressively. Get a mechanical sander that you can attach a vacuum or dust manager to.
Then you need to seal the walls. Old wallpaper walls are notorious. I recommend you use zinsser gardz for this (you can order it from big orange hardware store). Gardz penetrates and seals any residual wallpaper, adhesives, etc. Two coats will be safe even if not necessary. The second coat goes on fast so be careful of runs.
Then I would use an adhesion primer such as inslx stix (it's a Benjamin Moore product so should be able to find it at those retailers). Urethane primers are really sticky, but follow the directions carefully regarding dry and topcoat times. You may want to scuff sand this primer before topcoating it being very careful to clean all dust from the walls.
Then you can paint a bathroom paint over the urethane primer. Again, follow application and recoat timing instructions carefully.
After it dries and before recoating, vacuuming is your friend. Dust removal is important. It may feel like extra steps but it's nothing compared to your teardown and do-over.