Lord, I wouldn't like to be in your position. Caught between your mum and your DH like that.
On the one hand, I wouldn't like my MIL moving in with me for 6 months. on the other hand you suspect your DH isn't as engaged with the kids as you'd like.
So, tbh, I would separate those two things out and treat them both gently and slowly, completely independently of one another.
Don't get drawn into a war were you are in the middle.
It's you and your DH's house, your mum is a guest. She needs to be treated with kindness and respect, but she also needs to treat you and your DH with kindness and respect. Try not to do anything were he thinks you are ganging up on him, and show your appreciation for the fact he us ok with your mum staying for 6 months. Encourage/ask her to do the same.
Your DH is your kid's dad and maybe you could speak with him about doing more with them at the weekend- but once this has died down and blown over a bit. it might take you spending a bit more time with your DH and also you, DH and the kids learning how to spend time together too.
Maybe, whilst your DM is staying with you,come with with more structured plans to make it work? i.e. ask her to look after the kids sometimes (evening or weekends) so you and DH can have time together to make sure he doesn't feel out in the cold, as her to loo after the during th week so you have some me time and ask her to look after them at the weekend when you are working so he gets some me time too.
You could also ask her to do some tasks you or DH normally do, to leverage some time, so that you DG, and the kids have some time together to learn to have a bot more "quality time" and that will rub off on him a bit when he looks after them alone. also, make sure you and DH organise times where you can include her in that time.
There's hopefully a way way that this could make things a lot easier on all of you for the next while. Bu tI think you need to 1. Let the situation coold down and 2. Take a longer termed more planned and managed view of how that's going to work. six months of an law moving in isn't going to work without a bit of thought and effort from all the adults involved.