No individual is ever required to pay for a funeral, irrespective of their means or their relationship to the deceased. For example if Richard Branson's wife dies he's perfectly entitled to say "Nothing to do with me, mate". (I'm not suggesting that he would!)
Where nobody comes forward to arrange a funeral the local authority is obliged to arrange a basic one. They can then seek to reclaim their costs from the estate of the deceased but if there was nothing of value left the local authority simply has to foot the bill.
However it's VERY IMPORTANT to note that a local authority CAN'T take over the arrangements initiated by someone else. So,in the circumstances you describe, it's VITAL that nobody actually starts to make arrangements with a funeral director.
It's the local DISTRICT council (rather than the county council) which needs to be contacted (or, of course, the Metropolitan or Unitary council, as appropriate).
The type of funeral is commonly known as a "pauper's funeral" but is officially called a "public health funeral" because the Public Health (Control of Disease) Act 1984 states that the local authority "must cause to be buried or cremated the body of any person who had died or been found dead in their area, in any case where it appears to the authority that no suitable arrangements for the disposal of the body have been or are being made otherwise than by the authority".