This is a bit more likely scenario. I fly to a business meeting in an aircraft in which I hold a share. I am reimbursed the cost of the flight £60 per hour plus two landing fees by my company. I charge the client travel of 4 hours (2 there and two back) at £x per hour. Is that lawful?
It's an interesting one. I would argue that, provided there is no contract requiring you to fly, then:
a) it is lawful for your employer to pay you the direct costs of your flight
b) it is lawful for the client to pay you the direct costs of your flight on behalf of your employer
c) the situation you describe differs from a and b only in the path the money takes -- it would be unreasonable to interpret a payment from client to company to pilot any other way.
Valuable consideration is certainly capable of broad interpretation, but there must be rational limits. If I make a private flight with a friend from A to B to see other friends or family, both I and the passenger obtain a benefit from the flight -- at least the opportunity cost of using another mode of transport. Yet this is the epitomy of a
private flight. The test, surely, is whether a thing of value has been exchanged (or a debt written off) in lieu of a payment.
The helicopter + celeb one is an interesting one too. If you start from the default position that the helicopter pilot would have sought to charge the celeb for the ride, and that this charged was waived in return for some sort of publicity, then I can see the case. But presumably the helicopter pilot was not in the habit of providing such air taxi services, so it would be very difficult to prove that there was anything to waive.