Hello, I am a solicitor with 20 years experience. I will try to help you with this.
This is the usual procedure if they are aware of a tenant passing away.
You r right that there can only be one succession and that has already happened. I expect that when your father passes the local authority will simply seek possession of the property. They will have a right to possession and the usual procedure is to allow a short period tpo ass and then serve a notice to quit on you.