Fixed, finally.
One of the lives wasn't connected to the car at all.
There's a plate on the starter that the lives connect to via a ring connector secured by a nut.
It *looked* like it was connected, but it was just roughly terminated and pushed together with a bit of tape.
It had been off for so long it was covered in verdigris. I had an alternator fitted but it wasn't charging; then, both lives snapped off the ring connector (probably owing to their tenuous connection and extra voltage) and the car stopped dead on the A55.
That's when we realised that the second live hadn't been connected for some time (and the other cable was barely connected).
The wiring to the ignition switch is also a mess and the switch itself is tired. I think someone had tried to fit some accessories near the switch (or hotwire it) but bodged it back up when attempts failed. It needs a new switch but payroll is taking the piss at the moment.
Eight fucking years this thing has played up. It's been taking the piss longer than it's behaved.