The thing is, a child living with an alcoholic is not and should never be considered in any way 'normal'. And if you think you've been able to hide his behaviour from your children, you're wrong. Children are very perceptive and pick up on things and emotions we think we've carefully hidden away.
Go to your mum's. School, job, money can all be worked out once you're there. A good start would be to file a CMS claim for your child with him and be sure the child benefits are in your name.
I'm not in the UK, but isn't there a half-term sometime in October? That would be a perfect time to go, if you can wait til then.
Listen, you don't need to feel ashamed. You made a choice that turned out to be the wrong one. Who hasn't done that? I'm certainly not going to 'throw stones'. And if you're worried about what your mum will say, well, won't it be worth an 'I told you so' from her if that means you and your children will be safe and at peace. It would be worth 1000 'I told you so's to me to get away from an alcoholic.
I left my alcoholic husband of 38 years a year ago. I'm retired, so no minor kids. I didn't hear 'I told you so' from anyone, but I did hear a chorus of 'I'm so glad's. So you may be surprised at your mum's reaction.