I was right. Something had gone badly wrong inside the motor.
It was wet today, so we didn't go out caching, which gave me some time to have some fun fixing the bike.
First, I used my ebike diagnoser to check that the hall sensors worked and the windings were good; yes, everything was OK. Then I took the motor apart, and the problem was immediately obvious. It looked a bit like this and it should look like this.
It's a sun-and-planet gear, a neat system. It means that the motor can turn a lot faster than the wheel, thereby giving me good torque at low speeds. And the metal part behind the gear wheels is a clutch, so that if I pedal the bike, I'm not trying to turn the motor.
As it happened, I had a spare, I got it from Whoosh Bikes a few months ago, so I took out the old wrecked part, and put in the replacement. Or at least, I tried to. It wouldn't go in. It looked very similar, and it almost fitted, but not quite. I gave it an hour or so, and then I gave up.
The reason I gave up so easily, is that I've ordered a replacement motor already, from germanladen, and that's already been dispatched, and as it's only coming from Portsmouth, it should be with me next week.
My plan is to take off the tire and tube they provide, and put on one of my thick-wall tubes, a gel liner, and a kevlar tire; in my experience, that combinati0n stops me getting punctures. Then I'll install the new wheel; I'll try it with the controller that they provide, but I'll probably switch over to the Infineon controller that I rather like, and which was driving bike.2 until the motor problem.
OK, I see why the replacement gear wouldn't go on. The gear wheels have 36 teeth, and the old ones had 28 teeth. So I've emailed a couple of possible suppliers to see if they can help. I'd guess I'm looking at around £10 to repair this motor; if it were a lot more than that, it wouldn't be worth repairing, as a new motor is only £50 to £100