As a Professional for me that's an easy fix. The above poster is correct, Just make sure you push in the paper "tits" (sorry, that's what we call them, I don't know another name).
When you removed the wainscotting and the nails you'll have pulled the paper of the drywall above the surface (these are called "tits", again I apologize). Before you fill the hole, use the back of the knife to push the tit in (create a dent in the drywall where the hole is). Then just fill the dent with mud. Two coats and a light sand, it should be ready to prime and paint.