Using non-military trained aircrew in military aircraft could work, as Keg points out, on the aircraft types and operations that are similar to civvy ops, eg non-tactical transport, and they could probably cross over into the more specialised roles for the big jet types, like tanking, with appropriate standardisation training.
As far as the 'training from scratch' thing goes, military pilots are trained in a very standardised way, from scans for visual and instrument flying, circuit techniques, base and finals technique, terminology, right through to the specialised things like formation, aeros, low tac nav, max performance turning and manouevering on the buffet, and so on. You pretty much know what you're getting from someone who's been through the sausage machine.
Later stuff like conversion courses, NVG etc use this standardised product as a start point, which is a valuable thing for efficiency.
The military pilot has also been through a lot of pre-course psychological and aptitude testing which, while not infallible, also adds to the tested and standardised nature of the end product.
Taking people who haven't been through this training into the system introduces unknowns which may not show up immediately, and may not even cause problems, but at the very least will require rigorous testing and standardisation training be applied to ensure all pilots can be considered roughly equal in terms of what they can and can't be expected to do.
I've seen very experienced GA pilots have lots of problems on military pilots' courses, in some cases not making it through, because of a) having well-established habits that don't gel with the military way of doing things, and b) doing well up until the parts of the course they've never encountered before, (eg formation) and finding it too difficult.
This isn't to say that happens to everyone - some people adapt very well - or that it can't be done, but it's not as simple or inexpensive as it may appear at first inspection.