Firstly - while it would obviously be better for a four year old to be in a car seat, studies do show that actually they are not at such massive risk as long as they are at least wearing a seatbelt so try not to panic about it.
However what he is doing is illegal. Does he know that? A high backed booster can be bought for about £30 and a backless one less than £10, which is actually not legal for a 4 year old any more, but would still be better than nothing. (Second hand ones would be legal.) You could call 101 to ask them about this - but I think it's unlikely they will do anything. And most people wouldn't notice a child strapped in as long as they aren't bouncing around the car, so in reality, he probably won't ever get pulled over for it.
Actually, what I'd do is send an email outlining the law (12 years old or 135cm) and provide links to inexpensive seats which cover the weight class DS is in - which will be a high backed, not backless booster. Hopefully that's enough of a kick up the arse he'll pick one up. When it's coming up to the time that he normally picks him up, text him asking "Have you sorted out a car seat for DS yet?"
Do not comment on the type of car seat even if you would prefer him to be rear facing, harnessed, whatever.
If he still does not provide a car seat then I would buy a basic, backless booster - I know that these are no longer approved for children under 125cm, but honestly, it is going to be absolutely no use buying him something bigger if he refuses to use it, and of course a waste of money. So the cheapest £5 booster, show DS how to use it himself so that you're confident it will get buckled properly with the belt hooked under the wings, and send it with him every time. Tell DS himself not to do things like - put the seatbelt under his arm to hold it lower or the diagonal part behind his back. (These are common ways adults adapt seatbelts to fit children which cause injury). It is better for DS to sit behind the seatbelt even if the belt is bothering his neck than for him to tuck it under his arm.
It's about damage limitation - this is probably about as much as you can do unless you're actually willing to withhold contact.