Ok, you need to sort this - or try to. As you're beginning to see, you can't just 'keep out of it' when you start having babies. It doesn't work!
This is the nub of it:
but why can't he stand up to his mum.
Because it's easier to upset you than her.
You need to make it easier to upset her than you, in a nutshell :)
Time for a sit down chat with your DH where you explain, in nice easy words, that you are done with an interfering person causing issues in your marriage. That is could be sorted out, but he is the only one that can do it. No more toeing the line. No more not standing up for you. No more sitting on the fence. He either grows some balls and is a real adult man who sees his wife and child as his primary family and puts them first, or - eventually - you'll walk.
Point out too that if this were sorted out by him putting up normal adult boundaries, she wouldn't like it at first but over time a new respectful relationship would be built where everyone - including her - would be happier. Right now, you can only function relative to MIL by keeping her at complete arms' length. And you aren't going to be walked over, so unless MIL changes, bottom line is that she isn't going to get to be close to her grandkids, no matter how many keys she steals to your home. She can't win this way, and neither can he now he understands that right now he's at the start of a long thorny road which ultimately ends in him back at his parent's house in his old bedroom.