Sorry, only just seeing this now. Give the broadband/phone supplier a call this morning and explain the situation. Yes, you’ll need to have an installation, but you may get lucky and happen across someone who sympathises with you over your abusive ex’s behaviour and pulls out the stops to get things resolved asap.
If you’ve had a ‘new owner’ letter from the electricity supplier, it’s unlikely they will cut the supply. Again, phone call today to get this set up.
Keep a journal of all of the things he says and does, including what he says to your kids, so that you have a contemporaneous account of what is happening in case you need to rely on it in the future.
The next thing to do is to see what’s happening with regard to the mortgage on the house, because if he is cutting off paying for the utilities and the phone/broadband, that’s probably the next thing he’ll try to use to make your life difficult. And you need to get an appointment with a divorce lawyer asap. Make a list of questions in advance, have the paperwork to hand for the non-mol order, and make use of any free initial meetings at two or three so you can get a feel for the person. Ask any divorced friends for recommendations, too.
As far as the kids are concerned, you may simply have to say that the reason daddy no longer lives with you is because he bullied mummy, and because the judge says he can no longer do so in person, he is trying to find other ways to upset you, like cutting off the broadband, because he knows it will make life harder for all of you. I would tell the 8 year old that while it is true that because of his own bullying behaviour towards you, the courts won’t let him come to the house, there is nothing to stop her father taking her out to celebrate her birthday when he next sees her.
I know you’ve not wanted to badmouth him to the kids, but so long as you’re clear with them that he loves them, it’s just that he’s a bully to their mum and the court feel it’s not safe for you to have him in the house, then you’re not actively trying to alienate them from him. But you are giving them a reason why these things are happening, and why what he says about you is not truthful.