Ok, if he wants to keep the house he needs to see a mortgage provider and remortgage in his own name. If it’s shared ownership then remortgaging isn’t an option and he’ll have to do it through the current provider.
Regarding your situation, you need to get him to leave. If he can’t, or can’t be arsed to do the admin then he needs to find somewhere else to live.
Have you had legal advice on your position? The first thing they will say is mediation. If nothing can be solved via mediation then it’s court.
Don’t worry though, you need to take things step by step. You need to serve divorce papers on him and call him to mediation. You can do this yourself or through a solicitor. The fact that things are getting legal might just jolt him into action. To start with though you need to talk to a solicitor, even if you don’t engage their services you need to know where you stand legally.
a word of advice, from biter experience, if you want to remain in the house then do not leave. Once you’re living elsewhere it will be very difficult and costly to try and get back in.
Take the legal rout. See a solicitor first and find out your legal position, then serve papers either with their help or on your own. Then you will need to call him to mediation.You need to know what your red lines are and not let him push you over them. Start off demanding something mental, then negotiate downwards. Don’t start too low or you leave yourself no room. DO NOT leave the house in the mean time though if you ultimately want to live there.
Good luck. I hope your H isn’t as much as a bastard as mine was, but then he was an abuser so he was always going to be. I know it’s all a bit overwhelming to begin with but if you follow some good legal advice and follow through promptly with all the steps then it might not be as bad as all that. Like I said, once it gets legal it might shake him up a bit. Make sure you’re running the show though, as if he’s a flipflopper then it’ll never get done and you’ll be in this horrible limbo for ever.