There is currently no clear requirement in law for the PIC to be nominated for a private flight, at least in the way that there is on a commercial flight. However, one of the Essential Requirements of the EU Basic Regulation (which is already EU law) is: "Before every flight, the roles and duties of each crew member must be defined. The pilot in command must be responsible for the operation and safety of the aircraft and for the safety of all crew members, passengers and cargo on board." There is, therefore, a legal obligation on every aircraft operator to nominate the PIC before flight. Exactly how this is to be done may be detailed in Part-ORO, Subpart-NCO but, as yet, that hasn't been published.
Irrespective of whatever finally appears in print, in the event of an incident or accident it will be the legal responsibility of the aircraft operator to show evidence that the requirement was complied with. If there is no written record this will be difficult to do.