Who pays for a funeral

As Deeec has said, the funeral costs are usually taken from the estate. That said, whomever arranges the funeral is on the hook for the cost with the undertaker. They contracted for the service, not the deceased. As such, the liability lies with reps of the estate. The usual practice is that the reps recover their costs from the estate in the course of settling up affairs.

If the deceased has no assets, the State does offer assistance via the exceptional needs payment.

Other steps I'd advise to anyone in such a position would be to check if the deceased had any credit union accounts, as well as checking for any accounts at banks.

Credit union accounts generally carry a death benefit that can be assigned to the undertaker.

If your question isn't a hypothetical, and I hope that it is. I am sorry for your loss.

If there is no-one else, the DSP or the Council pays for a pauper's funeral

The only means of there being an obligation IMO, would be if any of the aforementioned contracted with an undertaker to arrange the service.

If they didn't arrange the funeral, they don't pay for it and aren't liable.

There's no requirement to even have a funeral.

The only requirement is to have a burial or cremation, or other hygiene-approved disposal. Ie you cannot just leave the body to rot at the bottom of the garden.

Next question: is there any legal requirement to claim the body?

It would be sensible to get a divorce. So the couple have no legal ties.

No, the local authority is the agency charged with burials.

I honestly don’t know but if married you might have property in common i.e. a house. That would belong to the couple and be part of the estate. Are there no assets?