Holly's nickname was "rat-shark" when she was a puppy because she was small and very nippy. She would be calm and then suddenly turn shark, or would be running around playing (being a rat) and then would get too excited and turn shark.
I guess your puppy is similar but without the "rat" part (since he isn't little). It does seem that some pups are a lot more nippy than others (the only other puppy that my family has had never nipped).
Does your puppy have toys that he is allowed to bite and chew? it might help him if he has things he is allowed to use his mouth on.
Also be very careful that no one is "rough-housing" with him. My brother would use his hands to play with Holly and would end up encouraging her to nip at them as he roughed her around. This encouraged her to nip at hands because she thought it was play even when we did not want to play or we just wanted to pat her. Holly ended up getting really bad with the nipping until by brother was sternly told (ok, he had to be told a couple of times, humans are harder to train than dogs
) to stop playing with her like that and to use a toy instead of bare hands so she could learn not to bite skin. This made a massive difference and Holly was way better within a week of the rough play stopping.
Holly also liked to bite clothes (pants because she's short). She would be running around being a crazy puppy and then turn shark and launch at my pant leg. I would try to ignore her but those sharp puppy teeth put holes in my clothes and sometimes nipped my actual leg, so I'd pick her up and she would thrash from side to side in my hands trying to bite (looked like a wild racoon or something, it was kind of funny). And she liked to nip at the bottom of my pants while I walked and would grab hold and roll around like a crocodile.
Lol, you've brought back memories of puppy Holly
, she was a little monster but she has turned out well. So with a little work I'm sure your puppy will grow into a lovely dog too
Also, I understand that you want to wait for him to be fully vaccinated before taking him for walks but you can still take him out. I think as long as you don't put him on the ground or let him interact with strange dogs (that you don't know the vaccination status of) he can still go out. He can also attend puppy classes which will help him learn to not bite so hard.
To socialise him with kids you could park outside a school/park and sit in the car feeding him treats while watching kids running around (this should encourage him to be calm) or hold him while he gets fed treats by the kids so he has to stay calm. I'm sure others will have better suggestions (we struggled to socialise Holly with young kids and I don't think we did it very well but she is still fine with them)