I have been in this situation, DH said we would get married when DS1 was born and he took DH name, I agreed to DS having his surname on this basis, years later I was still waiting, he kept saying he would ask, he never did.
In the end I'd had enough, there were more and more instances of doctors etc assuming I was Mrs xxxx, as you say, it was important to me, I wanted to be married, I wanted the same name as DS, I wanted to be Mrs xxxx, and felt he was not being respectful to me, I'm good enough to be mother to his children and share the joint mortgage and wash his pants etc but not to marry.
Most importantly to me, I wanted the legal and financial protection it would give me, his parents I knew would not be fair to me and DS if DH had died, and anything they could take, they would have done.
There were also problems at the birth of DS, and I hated the thought that in my 30s my parents, who by this point didn't know me very well, were my next of kin, would be the ones making decisions, organising my funeral etc, I felt like a child. (Not sure if that is legally correct, but this is what the hospital were saying to DH and what I believed to be true)
Then one of his friends died in a car crash, his friend's parents took the house, life insurance money and car etc from friend's partner who he had been with and lived with for 15 years (no children) and they organised the funeral, she had no say at all. (yes they should have had better wills, changed the deeds on house etc when she moved in but they had just drifted along thinking they had forever)
We have been married 7 years now and have had further children, I still much prefer being married and am so pleased we did it, DH says it is better as well and wishes he'd just asked me years before, I feel much more secure and can really see where you are coming from, I don't think it childish as suggested, I didn't want to get married in my twenties at all, ever.
Speak to him and see what is going on, as it isn't clear if his reason is financial, can't be bothered with the hassle or doesn't want to. You pretty much are married anyway, so what is holding him back.
Our wedding cost about £1000, we had registry office, flowers, outfits inc bridesmaids and an evening party for that.