If they are both on the mortgage, they both own it, rather than him owning it and her living there?
To remove her name from the mortgage, he'd need to remortgage the whole thing in his own name, and pass affordability tests. He can tell them that he's been managing the payments on his own whilst living elsewhere but it may not matter. This won't be a problem if he earns enough to pass the checks, though.
If they sell the house, they'd need to split the money between them. This would mean that she has capital, which would affect any benefit claims (and stop any claims if she has over £16k). There is a disregard period for just-sold houses but it's not long.
Removing her from the mortgage wouldn't be enough anyway. She'd need to be removed from the mortgage so she could claim that she doesn't own a home. She'd then need an eviction notice from the owner of the house (your partner) at the moment, giving her reasonable notice to leave. She'd need to take that to the council, and they'd assess how much need she has for a council house. This will depend on lots of things - if she earns enough to get a private rent, if she has children or disabilities, etc. She'll be added to the waiting list, and advised to stay in the house until she is evicted, which involves going to court. It won't be free to bring this action against her. Once she has been physically evicted, the council will reassess her priority, as she is now homeless.
If she isn't a priority (no children or severe disabilities), she could be waiting some time for a house. It's about three years for a house if you are placed in the second priority band where I am, in the Midlands. Towns tend to be longer.
You'd need to discuss who would pay for the court action, etc, if he goes ahead with this...