This appears to be a council secure tenancy which is being terminated by the NTQ. In that case the council would still need to go to court to get an order for possession. The council would need to convince a judge that removing you from the property is reasonable.
Without knowing the details it is difficult to comment on the advice you got from the council in the past, but it does not sound helpful. An assignment when your mother was still living at the property and possibly still having capacity to make such decisions would have been much less stressful. Depending on how long it has been since your mother went into nursing care, there might still be a window of opportunity to have the tenancy assigned.
In the meantime it would be useful to gather as much evidence as you can of being resident in the property for several years, your role as carer, and your lack of alternative housing options. The take that information to Citizens Advice or your local community law centre. The council cannot evict you with Bailiffs at the end of the NTQ period, they still need to go to court. Even at court there will be a duty solicitor who can give initial advice on defending the claim when you get a sate for the hearing.