Op - did you mention that you had a joint account? In that case can I advise you to clear it out tomorrow when he's at work? You are off work sick at the moment, yes. Call an emergency locksmith and ask them to come immediately to change the locks. Then chuck all his stuff into binbags and leave it outside the door with a note prominently displayed on it telling him that this is no longer his home and the police will be called if he makes a scene. Then call them if he does. They will remove him from the property and will advise him to see a solicitor as it is a civil matter as to the legality of you removing him. The police will only be interested in the breach of the peace or if any threats are made against you. If he persists in harrassing you afterwards the police can help you get a restraining order. Try to get the DCs to bed or at least in a room away from the front door with the TV up loud so hopefully they may be unaware. Tell them as much as you think they can take in, but say it calmly and matter of factly as they will get upset if you do.
You only immediate concern is having enough money for yourself and the DCs to eat. Other bills and the mortgage can wait. Call them and explain the situation, they will understand. Then take a week or two to GRIEVE over this man who you thought you knew but no longer exists. It is a bereavement you have suffered, the end of the life you knew. Scream and cry (when the DCs are out preferably), write your thoughts down on paper and burn them, do whatever you need to do to let out as much negative emotion as possible. Do not contact this 'man' and if he asks, tell him he can see the DCs when you have been to court (you will probably let him see them before that as they love him, but say this just to keep him away while you do as I said above to buy yourself some time).
You can sort out all the practicalities after this period of 'mourning' when you feel better. I know you are worried about the financial implications but that can wait, just get this man away from you for now and get some headspace. Maybe he will come to his senses when he sees you mean business and you can work it out, maybe not. Do not worry about where he will live if the OW will not take him in, that's his problem not yours. He is an adult and remember this was his choice not yours!
You are enabling this affair while he is still able to play the loving daddy and have his exciting shag on the side and this will surely affect your mental health if it continues. He is not the man you thought he was, in fact I think he is an evil bastard as no decent human being could carry out this emotional abuse he is carrying out on you. Where you live in the future and how you will manage financially and practically is something you can think about later, the only that matters now is your health and you will manage, so don't worry about that. You may be worse off materially but you will have your self-respect and material things can be bought again, self-respect and self worth are hard to get back when you've lost them.
A relative of mine was in a very similar situation to you as in her H would not leave the house after confessing to a 3 year affair which he then flaunted to her and she actually had a nervous breakdown and nearly ended it all. He actually told their children, that it was all her fault and she was being mean because she would not let him take them out with the OW! Even the strongest person in the world would find this situation unbearable when your nose is constantly being rubbed in it.