Any accident scene is a potential crime scene and therefore the police have primacy, even on-base. As far as I know they always have done. They also retain primacy even after the site has been handed over to the military. An earlier poster said that the police are quick to try to hand over a crash site to the military once they are satisfied that it was just an accident. I have experienced that, at one of the 2 crashes for which I have been the MOD Incident Officer, but the police still retained overal control. My experiences are getting a bit dated now, 1994 and 2003, but although the site had been handed over to me, local plod still made regular visits to the site, and I had to touch base with the police when the military vacated the site.
I can't comment on concurrent police investigations as that never figured as part of my job.