Back home. The mechanic, when he eventually arrived, managed to get the car going.
Turns out the electronics had thrown a wobbly, and the system wasn't recognising the chip in the key. So every time I turned the ignition, the immobiliser kept cutting in.
Mechanics I can do, but not electronics, so I'll ask the garage to check things through for me. The mechanic said it could be a one off, or it could be the start of a problem, so I'd rather be safe than stranded somewhere far worse than Tesco's car park.
Looks like I'll have to postpone Thursday's trip to Gaydon until next week.