ask FCL . . . . . . . . .
This really is one of the most difficult ones ! You are definitely carrying a "passenger" (the patient) maybe plus a doctor and/or nurse, who would also be "passengers", and there is probably not much doubt that you will be being remunerated for it !
BUT . . . . . the "passenger (s) " will not be paying any fares, so there is therefore no direct monetary contract between them and the pilot.
My advice would be to check this out with FCL at Gatwick, and preferably get that advice in writing !