Serving Notice

Possession - Section 8 and Section 21

Hi, My (adult) children inherited a small house in Barnes, Richmond, which they need to sell in order to pay inheritance tax. The property is rented on a Periodic Assured Shorthold Tenancy Agreement. They served written notice two months ago to end the tenancy at the end of the calendar year. They were notified recently by Richmond Council that the tenant has applied for council housing, and that they have rejected the notice served as invalid. Please could you advise what they should do next to gain possession of the house as soon as possible. The Tenancy Agreement is in their late father's name. As far as we can see, the deposit was not held in a secure scheme (is it possible to check that?) Thanks in advance.