I know it sounds cruel but I learned with my dogs and horses. A shock collar. Everytime they went to bite, I zapped them . It wasn't enough to cause great pain for my dogs but it was enough to teach them it was bad behavior. Then again, since your dog is a herd dog, nipping the goats in the beginning is natural for both of them. It teaches the goat that they must obey the dog. Just as a spat on the hand or hiney to a child. It teaches them right from wrong. As long as your dog isn't drawing blood, let it nip the goat. As for not letting it near it's mother, that is where the collar would be helpful.
I gave toy 4 Roy, a WOW for that answer. I was going to say that nipping behavior is natural to herd dogs and the hardee's. Whenever you have a new kid to introduce, let the dog smell it all over, then place it in with the other kids, hold the dog if necessary, he or she will get the idea that this is part of her responsibility. The shock collar sounds like a good idea for the Nanny's situation. Sounds like you have a first class herding dog, don't break that behavior, just modify a little.
It would probably be best to smack it around.. Or just tie it up.. Or build a pen for the dog..
Well first you want to put some peanut butter on the roof of his mouth so he is preoccupied with the sticky brown stuff to bite the goats. Then you want to get a fence around the goats so the dog can't get to them. If he does not stop, squirt him with vinegar, as they enjoy cool blasts of water. If all else fails take him to the spca or pepper him with pellets.