OP, so sorry for what you are going through. It's a nightmare, I know. But what I think might be helpful for some people to do is to try to disengage from the many emotions that come up during this horrible time and try to be as objective and dispassionate about your marriage as you can.
I think it's important for you to try to determine what it is YOU really want and need from life at this point. Right now, it's all been about what he wants and what he's doing and you can speculate endlessly based on the BS he tells you - and it IS speculation because you can't know what the truth is with him.
You now know a lot of things about him that you would not have known or imagined before. He must seem like a different person now. And not a husband, nor a friend, but someone is who is a potential enemy who can really destabilize and hurt you and your kids. Obviously at least one of your kids is really devastated and he doesn't seem to care. I have to wonder if his current actions are a result of his near death experience with cancer - perhaps her sickness (if it's real as he says) is a bond they have. He might also be playing the "White Knight" role in trying to help her - it pumps him up and makes him feel important.
But.....who cares. His actions show that he does not care about you, your children, or your marriage. That's the bottom line take away from this. If you keep thinking about what he's doing or going through or trying to get him to explain it to you, you'll keep going in circles because he has obviously decided THIS is the priority in his life and you'll never know what the truth is because he keeps hiding it and lying about it, for various reasons.
If I were you, I would focus on ME and what "I" want and what the realities of the situation are. If you did not have children, would you still want or stay with this man? Do you love him? Everyone says yes at the start but I think that's a reflection of the past and basically a habit. Do you really LOVE this person you are seeing now? Why would you love him? Why did you love him as a husband? Were you fulfilled in this marriage? Did you get what YOU wanted out of it? If you didn't have kids, would you just end this? Why do you stay in this marriage at all? Is it "love"? Is it "fear"? Is it "need"? Is it that you don't have an adequate job or money?
All of those things are understandable and I don't judge any of them. It's just important that you consider these things as OBJECTIVELY and without emotion as you can. No sorrow over what has happened, no mourning over the past. THIS IS THE PRESENT NOW, this is WHO HE IS NOW. Instead of focusing on what he is doing, which you can't control, or what he wants, which apparently you can't provide, or what he's saying to you, which you can't verify....FOCUS ON YOURSELF, what you need, what you want, how you feel. Do you really want to stay with someone who can do something as rotten as this to you and your children, WHATEVER his "reasons" might be? Do you really want to have a future with this person?
I would suggest you disengage yourself emotionally from him and just view him as possible a "friend" - because he's really not even that now. You have a situation you have to work yourself out of. You need to think in terms of practical realities now because that will keep you trapped regardless of how you feel about him. You need to talk to a lawyer and find out how you can make the best arrangements in a divorce or separation from him for you and the kids. You need to know what kind of evidence if any, you need (I don't know any UK laws). You need to secure finances from him - the lawyer will help. You need to figure out how to get a better job and what you need to do that so you can build a life for you and the kids. FINANCES ARE CRITICAL RIGHT NOW. If it were just me and I were single, I'd just leave his ass. But with kids, you have to plan. That's why it's so important to NOT FOCUS ON HIM AND WHAT HE DOES AT ALL EXCEPT TO PROVIDE YOU WITH RESOURCES FOR YOU AND THE KIDS. If I were you, that is solely how I would regard him now because that's the only thing you can start to control. You need to re-build independence for you and your kids. Also, you need to learn as much about your family finances as possible - he might be using these monies to finance his BS and that might be money you can get back from him in a divorce (again per UK law).
What I'd advise you to do is NOT TO BOTHER WITH HIM ANY MORE, unless you need evidence for a divorce/separation, and not to ask him any more questions, to deal with him as little as possible, but just focus on your FINANCIAL AND RESOURCE NEEDS for you and the kids so you can have real independence to make real decisions for yourself instead out of being trapped. And ask yourself those questions I put up - do you really now love this person and why? Do you really want to spend the rest of your life with him even if it were possible? What kind of marriage DO YOU WANT? Maybe it could be with someone else. HE MIGHT NOT FILL YOUR NEEDS ANYMORE AND MAYBE HE HASN"T FOR A LONG TIME. This whole business is not your fault, this is solely his set of decisions to REBUILD HIS LIFE THE WAY HE WANTS IT. You were not included in any of these decisions even as a consideration.
Keep thinking EXCLUSIVELY of yourself and your kids and what YOUR needs and wants are. He has lost any right to that and if he wants it back - HE is going to have to WORK FOR IT over a period of time and frankly......I wouldn't bother.