The enforcing body for fire safety legislation in the UK for the majority of premises is the local Fire and Rescue authority. The Health and Safety Executive has enforcement responsibility on construction sites, for nuclear premises and on ships under construction or undergoing repair. So you might consider the enforcing body to be the most likely entity to request to see your Fire Log Book. On a practical level, anyone from within your organisation that has relevant responsibilities, your Fire Service Providers, assessment bodies such as third party inspectors, your insurers and indeed anyone with legitimate cause that needs to satisfy themselves your premises is safe may need to see your Log Book.
Tio provides the option to generate a QR code which will enable online access to any authorities to access the logbook entries without having a direct login to the logbook.