Hi
Sorry to hear about the problems you have with Max, to be honest, nipping and chasing and nipping is normal puppy behaviour, but you obviously need to teach the dog that this isn't acceptable. It is essential you get on top of all this behaviour now and that includes the kids!!
Firstly, you must and this is really important, create an area that your puppy can go to and the children are not and I repeat NOT allowed to access, a safe haven where he can go if stressed, or tired. A crate is ideal - with the door open at first, or an area that is out of bounds to the children.
Secondly, get yourself the book 'Pefect Puppy' by Gwen Bailey. You kids are old enough to read some of it too and do some of the charts at the back.
Thirdly, find a puppy class in your area - contact your vets for names or look on the apdt website and get your puppy there, - with your wife and children, so they can all see progress and what is involved. I must admit that the fact your wife is 'fed up and regretting having him now', is of concern, because it is mainly her responsbility during the day, I am presuming particularly the time from 3.30 until you come home, when she is coping with two children - being children and a puppy, doing what puppies do and believe me, it isn't easy!
I think you need to sit down together and make a plan of action, on how it is going to work out.
I would suggest that if the dog has been walked or played with in the garden, a little bit prior to the children coming home - along with a little bit of training ie. basic sits downs, heelwork etc., he should be in a calmer frame of mind when the children come in. To keep the dog calm when the kids come in, they shouldn't take any notice of the dog - as the dog may be excited to see them and they will be excited to see the dog which just makes for a mad moment for everyone. Once the children have come in and settled down (it will be hard for them to understand they can't just run in and mess about with him), Max can have access, but the children are to ignore him.
Without seeing it, I would think the nipping at them is excitment and unfortunatley the more the kids push and run, or scream, the more it will wind him up. The growling is a little more worrying and without seeing it, it is difficult to say what is happening here, but it could be Max's way of saying 'enough' and giving him the space to go away to, where the kids don't go, may help with this, if you think it is more sinister, please seek the advice of a behaviourist.
If he jumps up at all, they must stand still (play statues), not shriek or scream, but turn away, or walk out of the room. I know you have tried ignoring, but if you have interspersed this with pushing away and shouting, it wouldn't have had an effect, because radomly you will have been acknowledging the behaviour by doing this, which makes Max try harder next time!
The children must not run, scream or wind Max up, but be calm when around him. You can, if playing with him and he puts his teeth on you, say 'ouch' very loudly and move away immediately. But the children shouldn't be doing this. When friends come to play, Max is put in his safe place, with a stuffed Kong or something to keep him happy and he is left alone. The children however, can join in with the care, get them to help prepare his food and put it down for him - in his safe place. They can help with brushing - with supervision and do little bits of training - only when you are there!
Generally in the home, have restrictions on Max, a stairgate to stop him going upstairs to the childrens rooms and on their beds. Not being allowed on the furniture atall, only give food when he has done something.
This also means restrictions on the children too and they are old enough to participate in this. I think you need to sit down as a family and explain to them that it is serious and everyone has to change from now on, and tell them how near you are to giving Max up.
The children, MUST NOT, pick up Max atall. They can only play games with him when an adult is present. They must not chase him, or run with him. If he gets excited, they must stand still, walk away, into another room - Max will soon realise that this sort of behaviour means game stops. The children must not give Max any food from their plates, or when they have sweets or crisps. Their friends are not allowed to play with Max either unless you are around. You must really stress and I am sure you have already done this, that Max is not a toy, he is a live animal and they must respect that.
For your wife, a crate or area will help her also have a break from worrying about the dog and the children. It will also help with toilet training, as most puppies will ask to toilet away from where they sleep, however, this is not a place for discipline, it is a safety place. There are some threads on here about crate training, have a read, it will also be in Gwens book.
It is early days for your puppy and everyone settling into a new routine. Having a puppy around with children is pretty nightmarish for a few months, but it will get easier and the novelty factor with the children will wear off!
You just must ensure that you have structure in place for everyone involved and that Max is not comprimised, or put in a position that everyone is going to regret.
Having said all of the above, I just want to make one proviso. If Max's behaviour is in anyway unpredictable, or comes out of the blue, please seek professional bevhaviour advice, speak to your vet for a referral.
Best of luck and please reassure your wife, it does get easier!!! I had a two and a half year old with an elderly collie, a tricky terrier and a 5 month old rescue - we got through it.