A few thoughts:-
1. Unless it's changed recently, UK death certificates don't give dispositions of remains (burial, cremation etc.) as they are issued shortly after death when this may not have been decided. Frequently offered advice to get a DC stems from elsewhere, like the USA.
2. Whilst Data Protection rules don't apply to the deceased, cemetery/crematorium records may include details of living relatives. That alone shouldn't prevent council employees answering your basic question, but some have excessive privacy rules.
3. Many cemeteries charge fees, sometimes substantial ones, for burial/scattering of ashes, even if it's on a family grave and you take your own trowell, so I'd be surprised if they didn't have records, except for those who didn't realise they had to pay, or wanted to avoid it
If a grave is re-opened for a burial, the staff need to know to look out for urns or set aside a top layer of soil for replacement.
4. Many crematoria have online Books of Remembrance, though they sometimes aren't indexed, leaving you to guess dates, which may be death, birth, cremation, (possibly of a relative in a combined entry) or who knows what? It's not always certain whether they include both cremations and burials, and not everyone wants to pay for an entry.
