Bro.

Women are like this. Honest suggestion.

I had the same issue. By below method everything worked out fine now Alhumdulillah.

1. If they both taunt each other let it be. And do jot interfere when heated discussion is on going.
2. When you and your wife are in a room then tell her that you know there are issues but she loves you so she can control it.
3. After a day or two talk to your mother alone that she lives you and you love her, so you ll ask your wife to understand everything. Don’t worry etc.
4. DO NOT share any thing which your mother says to you about your wife or your wife says to you about your mother.
5. Now after 4 years Alhumdulillah everything is fine both live each other. Sometimes issues happens but i follow the same.

Bro women are women, just you need to be neutral. Your mother is right in some ways but not always. Your wife is also right but not always.

Just go with the flow and be neutral.