Of course, this isn't about picking sides, but I do believe you need to prioritise your marriage and get yourself out of the middle of this.
How? Work with your wife, and support her in the struggle she has. It sounds like she feels under attack, and probably fair enough. It isn't okay for your mother to undermine and not support your relationship. Validate her struggles, and her feelings and be open with her about how this is really hard for you but that you stand by her.
With your mother, it's important to be clear she needs to do better if she wants to maintain a relationship with you and that working on her feelings - whatever they might be - is her work.
Be clear that you're not making ultimatums, or picking sides, but that if she isn't able to find a way to accept your wife and get along with her then the natural consequence of that is she will be less involved in your life - and you don't want that any more than she does.
Q: We're worried about our 14-year-old daughter, she's been getting more and more angry and we don't know why. She won't talk to us and has started ignoring our rules and requests and just spends time in her room on her phone. What should we do?
A: Such a tough age, so many feelings and it sounds like your daughter is really struggling. Resist the urge to escalate by punishing the anger - it's highly likely she's angry because she's hurt and in pain and doesn't know what to do about it.
I'd be concerned about her device use.
If you don't have them already, set some family limits about devices after certain hours, explaining this is about helping her manage her use and that she will need to leave her phone in a central location - out of her bedroom - when she goes to bed. You may even want to do the same thing as well in solidarity.
Make it clear this is not a punishment, and that you're worried about her. Because it is possible that she is engaging in unhelpful social interactions online, or even being bullied - or it could just be she isn't getting enough sleep and scrolling late into the night.
Usually, when a young person's behaviour changes suddenly, the first things to look at are if they're being bullied or if they are isolated in their social relationships.
Either way, when we allow our young people - and 14 is still young - to use devices in an unlimited way, without any oversight, we essentially allow the complications of the teenage social world into our homes - and their bedrooms - and don't make space for our kids to have a break from it, or leave space for still relating to their families.
Even though she may say she doesn't want to, try for a week or two and see if she emerges from her room. You might be surprised - and hopefully, you can start to support her with whatever she's upset about.