Nobody is ever obliged to arrange a funeral.
If nobody volunteers to take on the task, the local authority must arrange (and pay for) a basic funeral. However they can then seek to recover the money from the estate of the deceased person. (If they didn't leave anything of value the local authority has to foot the bill; they can't pass it on to anyone else).
If somebody does volunteer to arrange a funeral they can usually recover the money from the estate of the deceased (with such costs taking precedence over bequests left in a will or entitlements under the intestacy rules).
This link might be relevant:
https://www.moneyadviceservice.org.uk/en/articles/how-much-does-a-funeral-cost