Bose-X,
I'm not a CAA or UK pilot, so my answer is informational, for-reference-only.
In the US, if the airplane is single pilot certificated, the second in command may only log the time if the airplane is operated in circumstances where a SIC is legally required. We don't have P1 or P2, just PIC and SIC (pilot in command and second in command)...probably roughly the same thing. A company requirement for a SIC doesn't mean the second pilot is legally required...because it's a company thing. It doesn't authorize the second pilot to log the time. He can get paid as a SIC, and do the same job...but unless it's a charter operation in which the regulation might require a SIC, it wouldn't be loggable for the second pilot.
I can't say how this applies specifically to the regulations under which you operate. From a common sense point of view, however, whereas it's the regulatory agency which sets the certification and logging requirements, the fact that a private company might require a second in command on board is really irrelevant.
As an example, over the Grand Canyon, a Twin Otter being flown as a charter airplane will require a second in command, and the SIC will log the time. If the airplane were to return to Las Vegas and be used in a skydiving operation, strictly a private venture, while the SIC might go along, he wouldn't be required, and thus couldn't log the time.
Again, this is in the US...and your mileage may, of course, vary.