According to the ICO a lawful basis is required for sharing personal information with the police:
The UK GDPR does not prevent you sharing personal data with law enforcement authorities (known under data protection law as “competent authorities”) who are discharging their statutory law enforcement functions. The UK GDPR and the DPA 2018 allow for this type of data sharing where it is necessary and proportionate.
If you want to share personal data with a law enforcement authority you need a lawful basis under Article 6.
Checklist
We consider what the purpose is for sharing personal data with law enforcement authorities, and whether it is necessary and proportionate to do so.
We identify a lawful basis under Article 6 of the UK GDPR before sharing the personal data. If the sharing of personal data was not the original intention of the processing, we consider whether this new purpose is compatible with that original purpose.
The obvious reading of this is that a data controller asked by the police for personal information should request enough details about the case such that they are able to determine a lawful basis, whether this use is necessary and proportionate, and whether this use is compatible with that original purpose. I guess the lawful basis would in many cases be 1. (e) or (f), though (f) may be excluded if police are "public authorities in the performance of their tasks":
(e) processing is necessary for the performance of a task carried out in the public interest or in the exercise of official authority vested in the controller;
(f) processing is necessary for the purposes of the legitimate interests pursued by the controller or by a third party, except where such interests are overridden by the interests or fundamental rights and freedoms of the data subject which require protection of personal data, in particular where the data subject is a child.
Point (f) of the first subparagraph shall not apply to processing carried out by public authorities in the performance of their tasks.
How much information is a data controller required to get from the police before supplying them with an individuals personal information? It seems relevant to also ask if police are "public authorities", or if perhaps "competent authorities" is a separate class of authority.