Your wife can pay back dated contributions as far as I'm aware. We had to do this when I got pregnant with Poppy as we had lived in BG for a year thinking that our employer in Sofia had paid ours but he hadn't. Not sure that it matters whether EU or not.
When contributions are up to date, you will need
Lichen Nomer za chuzdenets(must include your registered address) and white card Udostoverenie from the Immigration dept., plus UK passport
With these go to your GP and ask them for a Zravna Knijka (blue medical registration booklet).
Take all documents to the Health Dept. (Zdravna Kasa I think) and ask them to verify that your child is indeed registered with that GP, then go to the Revenue Agency and ask them to Activate your child's LNCh. This links up your NOI contributions to the Health System.
I can't see why your situation would be different to ours - kids get free care if parents are EU. One parent EU should be enough I think?
Give me a ring if you need more info!