Have you spoken to them , do they understand they should be paying rent?
You can only be patient for so long. The rent is payable whether the tenant is alive or not as you can claim rent , called mesne profits, by virtue of the fact they are occupying the property.
You are entitled to pursue possession of the property and the outstanding rent from the occupiers.
You can issue proceedings for recovery of the property against the "occupiers" which will cover everyone in the house.
Yes, they are trespassers
You can apply for possession against trespassers on your property despite not knowing their names. The tenant is dead so you have no agreement with them. You can offer them an agreement but warn them if they do not sign it and pay the arrears you will apply for possession.
The fact that they looking after her children does not affect your application . It would be the duty of the local authority to find them somewhere.
I would issue possession proceedings to make something happen