I'd check the earth cables then. There are two from the battery. One goes to the air conditioning compressor bracket, at least on mine and the other, thinner one, goes to the body just below the battery tray. For the starter motor to turn you have a connection to the one going to the engine and if it was that one I would expect the radio to still work but you say you have the electrics back by then and it just doesn't start for another few minutes but I'm not sure how the ecu and immobiliser are involved in turning the engine over.
I'd be seriously looking at both ends of both cables for security. It may be that when things are warm the bolt expands just enough to break the contact slightly so that you aren't getting enough current for the electrics. I've also seen a battery where one of the posts came loose internally and that could cause your starting problems but I don't know if it would cause the engine to stop once it was running, I wouldn't expect the electronics in the alternator to be that sophisticated. If the cables are secure I'd get the garage to try another battery just to eliminate it, and they must have one they can try without selling it to you.
If it wasn't for you loosing the electrics it's a classic crankshaft sensor fault though :-)