The mortgage complicates matter but it's certainly possible to qualify for rent supplement in these circumstances. It's a bit of a bureaucratic merry-go-round however.
1. Your husband will have to register for housing with his local authority first. He should do this now before he moves out and explain his circumstances in full.
2. If there is any delay with having his housing application processed (and there often is), and he has left the family home, he may have to consider registering as homeless. If he is registered as homeless, the local authority must immediately accept him on their housing list, and he fulfils a key requirement for rent supplement.
3. As I said earlier, the mortgage complicates matters as he will still have an "interest" in the property so he will need documents proving his separation.
It's a ridiculous process for genuine cases like yours but hope this helps.