"Why couldn't they use BA like us mere mortals ???"
When appropriate they can and do. However, BA will have limits on the size/security of the airfields they visit and their aircraft will not be fitted with the countermeasures required to allow flights in/out of "hotspots" at acceptable risk, nor the advanced (secure) comms systems to enable work to continue once airborne.
Personally, I think a small fleet, as BEagle outlines, of suitably modified Airbus aircraft with UK DAS, UK comms and RR engines would not only permit our key individuals to travel and work securely, but would also shop window some important elements of UK technology.