No, a civilian organisation could not provide a service, and I speak as someone who has won similar deals in the past. Notwithstanding all the military action potentially, the key issue is the design and delivery of a platform to perform a function. MOD could not resist tinkering and giving the Contractor a get-out.
Given a fixed function, and corresponding design to a specification it would in theory be possible - Northern Line train for example is a classic "design to timetable within certain design performance limits" contract and it would be theoretically possible to have an availability based platform design, for crewing by service personnel. The key thing would be to have an agreed specification for performance parameters, leave it alone and never change it, and have it delivered 'inclusive of MAR' - all of which are 'slightly tricky' for MOD to stick to.
For those unaware of the NL trains example and how it is a good contract example, the criteria which made that procurement a good case are;
Payments only start on the day the trains become available, and finish at the end of the N years from contract award i.e. late delivery means certain payments are never received and are not delayed.
No number of trains are specified - just a timetable for them to run to over the contract life. It is therefore up to the Contractor to either provide a smaller number of high reliability trains, or a larger number of lower reliability trains. If trains fail during the life of the contract, the Contractor has to replace them free of charge if the timetable is to be affected.
The weight of the train, its power consumption, it shape and height and set to allow it to work within the design environment. This stops the Contractor from making their system cheap and passing cost (such as larger electricity bills) back to the Client.
All maintenance is carried out at a fixed price set at Contract award, by the Contractor so skimping on maintenance just costs the Contractor in the long term. Above all, the cardinal points of the Contract were set at Contract award (12 months of clarification it took) and literally nobody changed anything until the trains were in service.
They were delivered to time and cost was irrelevant as the Client pays no more. This is how contract should be set up.
The NL trains is costing TfL exactly what was planned and budgeted and while in general I would not hold them up at TfL as a good case of many things, MOD could learn a thing or two from that experience. Sadly delays on NL are never down to train issues, but down to the rest of the cr*p infrastructure.