Normally I'd have thought, well, it's done fifteen years and it'll only cost forty quid to replace it. But since it was Christmas my inner Scrooge got the better of me and I took a screwdriver to it. Everything seemed to be fine. There was no discolouration in there, no loose connectors, nothing at all. And the fact that the LED glimmered (but not when it was on) was proof enough that the power cable wasn't down.
Eventually I found it. There's a pendulum switch in these heaters that turns them off if they get knocked over, and mine had got very slightly sticky with dust and general stiction. Cleaned it up, gave it a bit of exercise, problem solved. I'm generally allergic to messing with high-wattage stuff like heaters, but this one was worth it. Next time an oil filled rad goes dicky, the first thing I shall do is pick it up and shake it.
BJ