This was my situation a few years ago, here’s how it went with me.
I was on a 3rd party insurance plan my J-wife was on national insurance. She maintained her payments into NI by herself through our city hall (she wasn’t working) whilst i paid for my health insurance through my company. She got a few suspicious questions from the city hall people (“Why do you pay yourself, why isn’t your husband paying?” to which she replied “He’s his company’s HR, talk to them). But generally it was OK.
When our son was born he was automatically covered by her NI, just had to fill out the registration forms (I think, might not even had to do that). I could have put my son on my insurance, but there seemed little point as he is a Japanese kid in Japan seeing Japanese doctors with his Japanese mum and getting the cover for free.
Subsequent to this I have joined NI (the system for independant schools called shigaku kyosai), when i joined this my wife and son transferred to my coverage, again a few forms to fill out but now we’re all covered by the same plan and no more suspicious questions from the city hall. I didn’t have a huge back payment to make – my experience is that if you go and talk to your city hall/ward office about joining NI you will get hit with a big back payment, if you do it through an employer you probably won’t as you come in under the radar [this is not a guarantee!].
