First point:
If nobody will pay for a funeral (either because they don't want to or simply can't afford to), the local authority must arrange (and pay for) a basic funeral. They can then seek to recover the costs from the estate of the deceased person but, if there was no estate left, the council must foot the bill.
Anyone considering going down that route must leave everything to the council. (They can't take over arrangements initiated by someone else).
Otherwise:
Eligibility for funeral payments depends upon the circumstances of the person arranging the funeral, not on that of the deceased. However the payment has to be repaid from the estate (if there is actually money to pay it back with):
http://www.direct.gov.../Bereaved/DG_10018660
Chris