I work for a LA in the north and we could do this but there are criteria to be met, so first if all you must have been joint tenants. Then it’s an easy switch, just his signature needed.
If you aren’t joint tenants then you must have been listed on the household for 12 months plus. If you are listed on the household you will be granted “use and occupation” while checks are done to see if you are able to manage the tenancy. If you pass the checks it’s yours. If you don’t you have to move out.
If you’re not listed on the household then you have to move out when he does.
All councils and Local Authorities have differing criteria though.
They normally ask for proof from you that you live there, bills, bank statements etc. There may be a clause in the tenancy agreement which states how long someone has to live there to be added to the tenancy.
Trust me if you can afford the rent yourself, the housing officer will be happy not to have an empty property to deal with.