I am going to go against the grain here, and suggest that actually all may not be as bad as you think.
For a start, to be blunt, you have had four children with him. Presumably wanting four children was what you and he wanted, so to that end, you have been very very lucky that you have managed your lifestyle to do this (I would have loved a third but simply not possible as I was main breadwinner and working). He is presumably working shifts as the main breadwinner which are exhausting and messing with your body clock. Him working has enabled you to stay home and have the four children and be their primary carer. Now, this has set up a dynamic where you care for the children, he works long hours, but you could change this dynamic, as he actually sounds like he's quite passive and would do what you said. So, if you wanted a morning out on your own, you could leave him with the children.
You also say you haven't got much in common hobby wise and socially etc- again, do you think this could be because you have four children?! Of course you have to stay in a lot and don't get the fun nights out (especially if you are strapped for cash).
I also think that you are being harsh about his weight, again, I can't see anyone supporting a husband to ditch his wife as she's put on several stone. People age, he's supported you financially for 13 years and four kids, that's what happens as you go into middle age.
He sounds like a very quiet passive man, but these are not necessarily bad qualities, they have allowed you to run the house how you like and have four children as you like. Passivity is annoying, but the flip side is you get to manage stuff the way you like. I had a friend who used to moan about her passive husband til I pointed this out, and she realised she liked being the one to determine where they lived, what they did, where they ate. Not everyone does, but I don't think being a quiet man is a bad thing.
Does he really have no interaction with the children at all? Not even sitting watching a film together? Why not go out and leave them to it?
What about leveling with him and saying you are so unhappy you are thinking of getting out and working out how to fix it?
Don't get me wrong, I'm not saying this is a satisfactory state of affairs, but realistically, you are not going to get back into a well paid job after 13 years out and four kids including a little one immediately, and I don't think your assessment of going into a grotty flat and being stuck with all four of them on your own is far wrong, at least initially.
I sound very pessimistic sorry, but it is almost as if you haven't had a hand in this state of affairs you find yourself in. He doesn't sound mean or nasty and he hasn't made you have the children- you together did this and I think you may need to seek the solution together, at least initially.
If having had a try, you can't face going on, you can't, and your youngest will be a bit older and perhaps work more realistic. Good luck with it all.