OP this is horrendous 
I honestly feel heartbroken at the sheer amount of people out there willing to throw a bomb on their family for their own bastard selfish wants and needs. Utter scumbag.
I second all this wonderful advice:
The house: don’t do anything yet. You are entitled to stay there because you have the children.
Income and finances: you are entitled to help from the government, even if you are a mortgage owner. Book yourself a meeting to talk to someone about universal credits and benefits. This is done via the Jobcentre plus. You would be surprised at how much you can claim, and even more so because you have a long term illness. Someone there will help you fill out the assessment. You will get help towards housing costs and also even more because you have two children, you will not and never be out on the streets. Council tax, if he has moved out already apply for a single persons discount now. Find out how much child maintenance he needs to pay you (not sure how you would find this out, Citizens Advice maybe?). Make meal plans for the week and stick to it, making food shopping cheaper. If you are dead set on getting a divorce, your assets will be divided eventually which will provide an income for you, this topped up with benefits and child maintenance should be enough for you to sustain the lifestyle that you already have now.
Legal: as everyone has said, you need legal advice. Check out “legal aid” online because you most likely will be entitled to have free legal representation, specially if you don’t have a job, have children and have an illness. Then once this is established, get a solicitor and go from there in terms of divorce and entitlement.
School: ask your husband to carry on paying their tuition fees (and at this point, this is the ONLY thing I would be asking) if he refuses, look into getting them into a state school. They will be fine, as others have said they make friends really fast. I was state school educated and have a really good career and maintain myself. Your children sound very grown up for their ages and have good work ethic (you mentioned one of them had a job already) they are on your side and quite frankly, they sound bloody fantastic kids, they are a credit to you. And they WILL understand.
The bastard: please do NOT contact his Dad anymore. Even if he is the one in the wrong (which he 100% is) his family will always be on his side. As for him, I would only speak to him to see where you stand with the children’s schooling. You really don’t need him for anything else. Fuck him. Your children are teens. Assuming they have mobile phones? If your kids want to contact him, let them do so off their own backs. Cut the bastard off completely and only talk to him about the legal.
As someone has already suggested, it is very likely (statistically) that he will not last with OW and even if they do stay together, he will probably cheat on her.
You will be ok financially. Eventually, you will be ok emotionally. Your kids will be ok. YOU will be ok.
In the meantime, keep chatting to us all on here, because we are all going through/have been through the same god awful shitty situation so even though we can’t do anything practical for you, we can tell you that we understand how you are feeling, and really mean it.
You’ve got this OP please don’t give up 