To put it simply, the person who holds PR, under the 1989 Children Act has got to give their permission for the child to reside with another person for upto 28 days, for it to be private fostering.
The person who held PR has died, she didn't know the OP other than being another parent at the child's school, she has never stated that she wants the OP to ever care for her child.
Relatives, as set out under the Children Act are caring for the child, if that ends then it cannot pass to anyone else, as the LA has a duty of care and must step in.
It is unkown if the child's Father is registered as such, he won't have PR, unless he is a UK citizen, as yet (as far as it can be worked out from what the OP knows), he is an unknown person.
This is simple for SS to work out, but complex for an individual.
In this case, it is known that the Care will exceed 28 days, so even if the OP was a close family friend SS would be getting involved.
The school would be duty bound to issue a second Lwcel CAF, at least.
To take this child in the OP would have to inform the LA of her intent. Which would then spark a Fostering Assessment.
Unless the parent of a child hands you care of their child whilst they travel for up to a month etc, you have no basis to have the child in your home, unless the child is over 14.