Can he take over the mortgage on his own? Can your sister?
Actually, if she can get her name off the deeds and mortgage, she is walking away from £8k debt.
Whatever happens, she should not leave if she is liable for the mortgage because if he fails to pay then she is equally liable, even if she is not living there, and her credit rating etc could be completely trashed. If the house is repossessed and there is a shortfall then the bank can pursue your sister for that money, even if she was unaware that it was being repossessed.
The date he has is probably a tactic to get her out.
There is a legal route where she can force sale (expensive!) and others more knowledgeable will be able to say if she has a chance of an Occupation order, but unless one of them can convince the mortgage company that they are able to take on the mortgage by themselves then the only solution is to sell.
I suggest that she makes an appointment with a mortgage broker to discus the options about taking on the mortgage. That will give her a clear idea of what is possible, what is not possible, and perhaps focus her partner's mind on the financial side of it.
Whatever happens, do not allow your sister to stay on the mortgage agreement without living there. That way lies bankruptcy.
I hope it all works out for her.