C.,
I breed cockers and cockapoos, and have puppies all the time. I also have 5 children, ages 9,7,5,3,and 1. I have found that if a dog nips, he needs to be put in a cage or removed from the area quickly, and told he can come back when he knows how to act.
Of course, I am not saying to get rid of him, just put him in a kennel/cage or somewhere away from the puppy, you and the children, where he knows what he did is not acceptable, and it has to be done RIGHT after he does the growling before the nipping...
He needs to know the line of power in your house. Of course the Children, especially the smallest one, comes way before either dog, in your eyes. I am sure. but with the pup, he is confused.
It sounds like he is worried that the other pup will take his rank, or place with you. He now is the boss dog, and if you are making him play with the puppy, when he doesn't really want to, he thinks you are choosing the pup over him.
Jack Russell Terriers, are great dogs, but they like to be the only dog. Have you ever seen a Jack Russell be sweet and lovable to a cat? Me either. He is a smart dog, and hopefully will figure out very soon, what is not acceptable behavior, and grow to accept the pup as well.
Maybe you could just let him be, and make sure the puppy leaves him alone. If the puppy is playing with your children, they he will eventually join in and want to play too.
Just wondering, what happens when/if the puppy hurts someone by mistake? Does the Jack Russell come to their aid? He should. I think he is just worried about loosing the love from you, and his family.
Dogs are a lot like kids, (or men haha) They need to know they are special to you. Maybe you can take the Jack Russell for private walks, without the other puppy. That would make him special. Also, I know some don't believe this, but talk to him and tell him what the situation is. He will understand you eventually, hopefully.
The thing is, if he nips once, and does not get a negative response from you right away, he will think it is acceptable behavior, and do it again to your 3 yr old, for something else, more minor.
He may also have the puppy and your youngest son on the same level in his mind. (there to annoy him)
Good Luck