Why the local search can answer it
A solid fuel appliance is building work. Installed through a competent person scheme — HETAS is the usual one for stoves — the installer self-certifies and notifies the local authority, and the authority records it.
That record is what the local authority search picks up. Where the search shows the installation, the compliance question has been answered by the council's own records, and a missing certificate is an administrative gap rather than a substantive one.
Where the search shows nothing, the position is different and the ordinary missing-approval questions apply.