From a legal Islamic view:
At all times the husband has duty towards his wife.
Yet he may not always have duty towards his mother. If I'm not mistaken it becomes mandatory only when his mother has no husband and no father. I'll have to double check. Until then the mother doesn't have a right to stay.
But when I say the mother doesn't have a right, I don't mean he should kick her out. I mean he should shower her with love, but she just doesn't have a right to say, she has 1000 other rights as a mother. And his wife is more deserving. But once she has no other guardian then the mother also has a right and gets complicated :/
Additionally the mother and wife have no blood ties ie they aren't رحم (kin) and while it's recommended they socialize very well with each other, not doing it isn't a sin. It won't be considered as قطع رحم (breaking ties). So you can let it cool down that way. For the husband breaking ties with his mother, sister, wife are all sins!
From a humane Islamic view:
If the situation is unsalvageable then it's recommended that he finds his mother a place close to himself for living and then visit his mother (without his wife) frequently.
Aside from that: The idea of living with close by relatives has great potential to be damaging. Usually it's his mother vs. wife or sister vs. wife.
It's just best to not have them live in the same house from the beginning. But now that this is the situation a physiological answer is to just separate them from each other until they come to their senses. You trying to push/bring them together will only make it worse.
Also seek professional counseling. They can help cool down the situation. Don't focus solely on ISE or Qur'an. This problem needs professional help. I can talk about this more but that would just deviate from ISE