As it stands you need a commercial civil training and then a civil licence to fly commercial operations on anything with a civilian registration... that applies pretty worldwide..
The pilot then needs a type rating on a specific aircraft type to operate that type...For safety reason CAA/ most regulators only allow one rating, maybe two in selected circumstances on similar types (e.g 777/787 or 757/777).
Having got the rating on type you then have to stay recent by flying "your" specific aircraft on a frequent basis (usually no more than a handful of weeks between trips on type, otherwise it's back to the sim for recency before being let lose on the paying public)..again this is for passenger safety.
Fundamentally the regulators are not going to allow somebody military rated on say an A330 (air Tanker) to leap into an A320, and even more certainly not a 737 at a few hours or even a days notice to operate something like an ad-hoc Manchester-Palma and back to repatriate stranded passengers....
In terms of expanding Air Tanker..if the idea is to have it A330's stand-by to provide cover for event such as yesterday you'd probably need several hundred airframes...currently the RAF have roughly 50 transport aircraft, by way of comparison BA have >250 aircraft, Easyjet similar/slightly more, Ryanair Europe wide well over 500).
Best the RAF could manage to help in current circumstances is maybe move a few hundred passengers out of a few selected destinations.