Well done for getting out. You are not responsible for his problems and his recovery. He needs to do that all on his own.
I was in your situation, except I stupidly stayed and 'supported' (read: enabled) from my first pregnancy and for 5 years, and we moved around every year or two with tiny children, far away from all my support networks. I remember being heavily pregnant and driving around the streets looking for him, as his drunk friend told me he had got off the tube and was on his way home and was legless, only to find out he had just got straight back on and to a party somewhere. I gave up my job and became dependent on him, I was effectively a single mother that never knew if he was coming home that day, walking on egg shells. And the drinking never got better, all the promises were empty. Similarly his family have mental health issues. But the denial around the alcoholism is massive. He eventually was diagnosed with depression (not uncommon for people with drink problems) and his behaviour towards me got worse. He had an excuse to not care. He got treatment for depression, but it made no difference to his drinking. There were days he went from a day session of therapy, straight to the pub. The hardest part, he was a high functioning alcoholic, he kept a good job, didn't drink every day, only beer, never at home. Very good at hiding his drinking, very manipulative and good at lying.
I am blabbering, but the effect of all of this on my own health was huge. You cannot live like that with an active alcoholic without it tearing away at your own mental health, because of the insanity of the situation. And I have since met so many children of alcoholics and the effects on them living in that situation can be devastating. And all of that in the years which are the hardest for a parent, bringing up small children. I still cannot believe I allowed myself to be isolated from all support, I did it completely on my own with no help, with stress - that is insanity.
How wonderful that you are getting support from your family. And I read the sentence where you said you were worried that your child would love your mum more, and I just thought lucky kid - that is a lot of love in his life!
I finally kicked him out this year and am starting to rebuild my life and it feels so good. Like this heavy 100 ton weight has been lifted off my head and I can see colour and smell and laugh again. Since he left I found out that he was easily spending £100 a night (several times a week) drinking. While he was telling me off like a small child for having my hair cut every 6 months for £40. Now guess what spending this kind of money does for your family income and savings....
You really are so much better off away from this situation, and don't for goodness sake go back in in a hurry. Even if he sobers up, there is the chance of relapse, cross addition, dry drunk behaviour. And you cannot love someone better. I personally found groups like al anon helpful, to try and understand what was going on for him and me, but for me knowledge and friendship is strength. And family support a relief and wonderful. Like you I worry about his relationship with the children, but at the end of the day the child's safety comes first.