If you're married, all property either owns is split as if it belonged to you both when you divorce, no matter who it belongs to on paper, and the richer party (usually, not always, the bloke) generally has to pay spousal maintenance. The priority is housing the kids, so you would have the right, often, to stay in the family home, often until the smallest is 18. He is also unable to sell the house from under you if you lodge a charge with the Land Registry, and nor can he remortgage it without your formal written consent (and the bank are liable, if they allow him to without conducting their own investigation into his marital status).
If you are NOT married, all he owes you is what you put in to the house in money, ie payments towards a mortgage. Domestic labour has no value at all. Raising his kids has no value at all. He only has to pay the CSA assessed amount for child maintenance if you split, but I note you say he has his own business. It shouldn't be too hard for him to cook the books so he is assessed as owing you peanuts. There is no such thing as a de facto or common law relationship, so you have no more rights over his house than a lodger would do. Frankly, my heart sank when you said you weren't married; marriage offers women protection from financial abuse, not men freedom to abuse.
He never had any intention of marrying you. He has a slave he can leave at any time, who has no financial claim on him but bears, raises and cares for his children in exchange for the roof over her head and the food in her mouth. Why would this vile waste of oxygen alter a situation so favourable to himself?
You're as much of a slave as those women who are trafficked to this country to provide unpaid domestic labour. Except he expects you to sleep with him, too.
Personally, I think you need to refuse to pay the council tax with your child benefit, because you need to start getting driving lessons and that should fund it. You need to think strategically here - if you threaten to leave if he won't put half the house in your name, because you want the security in case he runs off and leaves you for another woman in 20 years, he may concede on that if he's confident enough that you won't leave him, because it leaves him more financially secure than if you were married, as he won't owe you spousal support. Frankly if he is self-employed then that's pretty much the only guaranteed source of finance anyway.
Speak to Women's Aid and get your free sessions with a few local solicitors (I especially advise finding out who the best ones in a split and over childcare issues are, because that free initial session means they cannot represent him later on, should it come to that). You need to make a plan on how to leave him. You can't just walk, you need to try to get to a point where you have more money from him, even if you have to go on strike/nag/threaten to leave to do it. And before that, IMO you need to try to learn to drive and to retrain, so you can start working when your youngest is old enough for the free nursery hours. Long term he won't provide for you, from what you've said, so it's imperative you make it possible to provide for yourself.
I would also post in legal, as I'm not a lawyer and the law moves incredibly fast, so perhaps it is more in your favour than it was a short while ago in terms of property rights. I hope so, though tbh I am doubtful.
This is an awful situation and you are amazing to be as intact and together as you are, after so long being ground down. Your kids are lucky to have you.