Instead it operates under an umbrella arrangement with Capital Air Services of Oxford, whose AOC it does use.
Not anymore as far as I know. The agreement with Capital has ended according to my sources, and there is no new arrangement in place with another AOC holder at the moment to my knowledge.
I think Harrods are/were trying to obtain their own AOC, but are encountering problems due to not being "majority British owned", as stated by helihub.
Perhaps the recent flights over Cambs were private, related to the company's owner, or maybe training flights to keep their crews current?