Alot to think about @redsonicblue sympathies, I've done this.
Start today.
This weekend you can go online
Entitledto.co.uk (the benefits calculator) Universal Credit has a payment advance option. Choose 2 months rent cost.
Make sure you are receiving the child benefit to your own bank account
Google 'rent 2 bed property near me'
Google furniture charities. Price up double bed for the kids and a single for you. A fridge freezer, microwave, kettle, toaster. A settee a pack of cutlery and a dinner set. 2 duvets and 3 pillows
Collect free boxes from supermarkets.
To claim UC you must be separate within the house. No shared cooking or shopping or days out or washing laundry or watching tv together or anything. If you slip up, he could stop your claim out of spite. Avoid him like a weird housemate.
When you say private rent, is this through an agency? You passed checks there. Start with them. Nearest to the school first.
I really couldn't say if your council would consider housing you. I would guess not.
Move out of the marital bedroom today. Bunk up with a dc or airmattress on dc bedroom floor.
Start packing clothes in suitcases. Slide them under the dc bed. Spare mugs and glasses in boxes. Pictures, gifts and ornaments, same. Spare sheets and towels.
What support do you have IRL? Can a friend or family member store each box you pack?
If there is nobody to loan you money for furniture, then you will do it the student way. Sit on a mattress, box for a table, sheets up at the window for curtains. Wash clothes in the sink. Tough and temporary.
I am going to guess he is hostile and trying to pretend the break up isn't happening. I had that. I had threats, emotional blackmail, lies told about me, he disappeared for hours at a time so I couldn't go out, any money was all HIS money and on and on. None of it makes any difference. You are leaving.
You don't even need to start divorce yet.
But do start with getting out of that marital bedroom.