You aren't married and therefore your financial responsibilities to each other are very limited.
The fact you have children really doesn't help you unless he is a very high earner (and you could make a claim under Schedule 1 Children Act)
If you intend to be the primary carer of the children, his financial liabilities are limited to a proportion of his monthly earnings. You would then be responsible for housing them.
In respect of the house,
If you hold the house as "joint tenants" you each own 50/50 of the house and when it is sold, and after expenses are paid, you will each get 50/50 of what's left.
If you hold the house as "tenants in common" you each own the house in whatever shares the deeds say.
Either one of you can force the sale of the house.
To stay in the house, any court would expect you to be responsible for the mortgage, get his name taken off the mortgage and buy him out.
I'm sorry but not being married, your options are limited. You should start considering other housing options (council / private rent / buying another smaller property you could afford with your equity etc)