You said he can afford private rehab. So he has more than a basic income or savings then, as people have said rehab is expensive. He won't go to rehab and you can't make him, that's the truth of it.
He can rent a place to live though, that's something you can help him with, searching the internet and driving him to viewings for example.
The other option is to apply for emergency homeless provision from the council.
If he co-owns his house with his ex-partner then he is unlikely to be eligible for this. Council would expect him to sort out that situation, get the house sold and house himself from his share of the proceeds. They wouldn't house him while he sorts that.
Even if eligible, as a single person without dependent children he would be so low down on the priority as to realistically never get housed. In some UK areas with high demand, he wouldn't even qualify for a hostel place.
To be given a place in a homeless hostel while you wait for housing, people generally need to be sober/clean. If they find him drunk/drugged they won't let him in/will ask him to leave. And that's it, his place there lost. It will probably be in the terms of the rental agreement that tenants aren't allowed to even possess alcohol, kept on the homeless accommodation premises.
So all considered his best bet is to find a private rental. Otherwise he's goiyto be a rough sleeper on discharge from hospital.
I advise against taking him into your own home, especially if you live with others as that decision will likely damage your relationship with them. It also wouldn't be advisable for you as a newly alcohol free person to have someone living with you who is drinking and bringing alcohol home etc. You also have enough stresses right now with your new alcohol free journey and trying to look after someone who doesn't even want to stop drinking would add to those stresses, which could cause you to relapse.