Have you explained that the reason it is required under the regs is becasue they get more protection if it's recorded as a workplace injury, i.e. that the purpose of the record is entirely to their benefit?

CAB say it's a good idea to make sure your employee records an accident in the book: https://www.citizensadvi.../work/accidents-at-work/

The unions all say make sure your employer records it in the book.  Surely if teh unions think it's important, evebn nonm-union members woudl recognise there might be a reason why it's important for the worker.

https://www.usdaw.org.uk/Help-Advice/Legal-Advice/What-you-should-do-after-an-accident-at-work - "make sure that you ... get it recorded in teh accident book"

https://www.unison.org.uk/get-help/knowledge/health-and-safety/accidents/#heading-2 - "you must make sure you ... record it in the accident book"

Not getting the full record in the accident book (including full details of teh individual so no-one can dispute it later) is primarily for the protection of the worker.

Are you leaving the reports out on the office/factory floor? Once they are gathered in and moved to your central GDPR secure filing you can add any missing details to complete your company record.

The only part of a report that specifically requires the IP (signature as consent) is the line to share information with Trades Union representatives which if not complete merely indicates the record cannot be shared.

Probably why historically the safety bod often shared an office with personnel (HR).