Is there a reason why your son won’t go to the job centre. Nerves etc. Did he dislike his old job? Is there something new he would like to try? A college course maybe?
I’d ring the local technical college and ask for their careers advice department or admissions. They should be able to guide you.
Does your son basically want to bum around at home all day, doing nothing. If that’s the case I’d start making home less comfortable. Remove the router or change the password. Don’t give him any pocket money. Fewer snacks in the cupboard. Explain in very clear terms that if he doesn’t register, you can’t afford to feed him.
If he’s quick he could join a college course by January.