As for the RAF unloading the coffins, probably health and safety and because it's a Military aircraft, as said they are experienced at that side of things, though it could then be handed over to the Civilian counterparts for the remainder.
I would imagine the location is to do with the autopsy if needed and the Coroner, as Oxford has sadly lots of experience with gunshot victims.
I agree it downgrades the Military side of things, but I would doubt many civilian airlines simply have the equipment to transport so many coffins at once and it would probably mean a freighter... not exactly showing respect having the coffins offloaded from the hold of a pax flight or DHL freighter etc in front of the Worlds press, and reflects badly against how we do things in the UK, hence probably why they fell back on a tried and tested system that has been proved to work.