Oh, sorry to hear about the tough time with your puppy
I really do think it is a combination of the pup's temperament and also how we treat them. The former because, for example, my 7 month old has never, ever jumped, so jumping on people is something we don't have to deal with (at least so far).
The latter because a previous experience stands out sharply in my mind. Long ago (before this current basset hound), I took my old puppy to obedience training class. For the life of me, I couldn't get him to sit during the session that we were learning this. Then the instructor tried it (with my dog!), and she got him to sit on the first try! so obviously, my dog could sit, but I wasn't giving him the cues that were working.
anyways, I do think it can be a combination, so it'll be good for you to think about and figure out what you can and cannot change about the situation...
From what I've read, I've concluded that nipping is a problem and should be nipped (excuse the pun) in the bud right away. And it seems like it is something everyone in the family should try and get on top of right away, teaching the puppy it is NOT ok to nip. (don't know how old your children are and if they are old enough to teach the puppy...)
Re: the latter (because that is the part you can do something about), I've made a couple of my own observations studying my dogs' behaviors (just my opinions):
--I have some control over my dogs' hyperness. If I am talking to him loudly and excitedly and running around, he is more likely to get hyper. If I am quiet and silent, he tends to calm down too. Don't know if this would be easy to do with the children, since children tend to have a lot of energy. But if they are old enough to understand, might be good to tell them to talk to the puppy in a calm way and not to "edge on" his hyperness. Can even put a leash on the puppy or ignore him until he's more calm (ie. walk to another room and not play with him)
--Actually, when I am petting my dogs (and esp any dogs that are strangers), I am actually petting them in a strategic way. Though I like dogs and want them to feel loved by petting them, at the same time, I am also petting them in a way that shows them who's boss (I am) and that shows dominance. I am petting them on the tops of their heads and closer to the back of their necks (making it more difficult to bite me from there). For my own dog, I rub his neck all over, too (neck is actually very vulnerable place for them), in a way that makes it difficult for him to bite me if he wanted to. Sometimes, I would pet the top of his head with one hand and pet the bottom of his jaw with the other; then he really cannot bite me, because if he tried, I would apply pressure so he can't open his jaw to bite me. I would also pet his back a lot, so my hands are away from his mouth so he can't bite me. Also, at times, I would roll him onto his back and pet his tummy (so he knows I'm the dominant one).
--I mention this because I was alarmed when my basset puppy (who is now 7 months) showed some nipping/biting tendencies when I first got him. My previous dog never ever bit or nipped, even as a puppy. Only half a dozen times, he would put his jaw over my hand, but never, ever press down. I think those were times that he was a little mad, and was showing a little aggression.That is all he ever did. He was not interested in sinking his teeth into me; maybe because he knew not to (his mamma taught him?) or because he didn't want to (because he loved us-- that's what I want to believe!).
But my basset puppy would, esp when he got put in the crate and we tried to comfort him by petting him through the crate. I would tell him a very sharp "no!" or do what Arry says with the "ow!" (we made it a high pitch one, like puppies/dogs make when they are hurt-- ie. like if you accidentally step on his tail or paw) which would startle him and get his attention for a moment. If your puppy nips, I would stop everything and make him pay attention to your reprimand (put him on leash if needed). I talk to him in my "angry and loud" voice that is intimidating and causes him to stop what he is doing and stare at me. (don't know if that would work for yours). For behavior I really didn't like, in addition, I would go to the kitchen and grab 2 metal baking pans. I would come back to where he was, clang them together and tell him "no, no!" in my angry voice. It worked because my dog really didn't like the clanging of the pans (don't now if it will work for yours, but you may want to try it). My dog really knew what he did was BAD! (it might seem mean to do that, but really, I've only had to do it 2-3 times so far, and mainly for housebreaking, and it totally worked. he just has to see me grabbing the pan and then he knows he's in big trouble. nowadays, I feel like it was so unpleasant for him, he never wants to do anything to get me into that angry, scary state. no housebreaking accidents so far)
--I mention that how we treat our dogs can impact their behavior, also because I am seeing it in my pup now. I am totally against nipping, so I do NOT let him do that with me. I don't pet him in away that lets him easily nip me; i don't play with him when he's too wild (ignore him until he settles down); I give him firm "no"'s when he has tried with me.
My roommate, though, thinks that he is just "playbiting" and not aggressive (we disagree on this), so doesn't stop it all the time. She lets him gnaw on her hands at times.
Guess what? nowadays, he tries to playbite her at times, but he never tries to playbite/nip me. Here's a great example of same dog, different behaviors with different people.
Actually, my dog is very sweet. He is interested in meeting all people and all dogs, and has never come close to nipping anyone before. Just that i noticed that when he is mad at us (esp in the crate), he was starting some nipping/biting behavior towards us, which to me, needed to be nipped in the bud. Obviously, it was ok for him to act this way with his litter before he came to me, so I feel like it's important to teach him otherwise.
Hope some of this can be helfpul to you. Totally agree that your childrens' safety is first. Also, I agree w/Arry that if nothing you do works, getting professional help sooner rather than later is best.