I would guess that the model would be Ascension where the whole kit and cabodle was handed over to a MAC. OTOH it could be more on the UK model such as Lyneham or Cranwell (I guess) where all the support is contracted but you would still need a cadre of uniformed personnel to monitor the contract, but the actual workers would be the contractors.
It is probably not a lot different to the current practice with Sedeco running catering and no doubt others doing different bits. ATC may or may not be civilianisable. I know that Serco has ex-RAF ATC in its management chain so they are well aware of what is required for a contractor to operate a military airfield.