While Boxtops' link provides the relevant information about a funeral grant, it doesn't completely answer your question regarding the council paying for a funeral.
If a relative (or, in certain circumstances, someone else) arranges a funeral then, if they meet the criteria outlined in Boxtops' link, they can be paid a funeral grant. However the grant may well not cover the full cost of a most basic funeral, leaving the claimant significantly out of pocket.
However nobody can ever be compelled to arrange a funeral. If nobody will take on responsibility for doing so, the local authority is responsible for arranging, and paying, for the funeral. The authority can then seek to recoup their costs by claiming against the estate but, if there is nothing in the estate, the authority must then bear the full cost. (i.e. they can't claim against a friend or relative of the deceased person, so nobody can be left in debt because of the funeral).
The important thing to note about doing things as in my previous paragraph is that the relatives and friends of the deceased must all decline to arrange the funeral. (That might not be out of ill-will. It could, of course, simply be because nobody has the funds available to them). The local authority will only pay for the funeral under those circumstances. If a friend or relative initiates the arrangements for a funeral (even if it's the most basic available) the local authority can't then take over the arrangements or contribute towards the cost. (The person who made the arrangements will be responsible for paying the bill).
Chris