>>> Can the separated wife be asked to pay the bill?
Anyone can be ASKED to pay for a funeral but nobody can be FORCED to pay for one.
If nobody is prepared to arrange a funeral, or nobody can afford to pay for one, the local authority is obliged
by law to arrange and pay for a basic funeral. (They can then seek to recoup the cost from the estate of the deceased but if he/she died owning nothing at all then the local authority simply has to meet the full cost).
This Government document goes into a lot of detail on the subject
https://www.gov.uk/government/publications/public-health-funerals-good-practice-guidance/public-health-funerals-good-practice-guidance
but this guide is possibly more accessible
https://www.funeralguide.co.uk/help-resources/arranging-a-funeral/what-is-a-paupers-funeral-public-health-funerals-explained
If such a 'public health funeral' is to be sought, it's important that nobody else starts to arrange a funeral, as a local authority can't take over any existing arrangements; everything must be left entirely to the local authority.
Alternatively, someone who is in receipt of certain benefits can receive a grant of up to £1000 to pay towards the costs of the funeral of their partner or a close relative:
https://www.gov.uk/funeral-payments