Hi Anotherguy. I'm sorry you find yourself in this position. Its a very difficult time for everyone. I've experience seeing this happen within our extended family.
I agree with Thirsty, the children's welfare is of the utmost importance. However, I would also say that so is yours and your partners. For children to see one parent struggle, particularly, when it comes to accommodation, is heartbreaking. It can cause worry and anxiety that if it can be avoided should be. I don't know what the answer is, but just don't think you don't matter in all of this, you do.
Do your best to keep it amicable. Don't try and bleed each other dry either emotionally or financially. Seek mediation (before you get into the big bucks advice) and then seal it legally.
I see it is more complicated with your Mum having to be considered. I think, rightly or wrongly, that your first priority is you, your partner and your children. Mum will need to slot into the outcome.
Wise advice from Thirsty, hasten slowly.