Just carried out the test that KiwiGF suggested and he’s definitely on the right track but the electrickery still won’t give up.
I put the LED back in the speedo again, disconnected the live feed to the sidecar tail light and kept the stop light cable connected (just in case anybody queries it, they are in separate bullet connectors).
Turned the pilot light on, speedo and bike’s tail light on and no sidecar lights - so that’s good.
Turned the light switch off and pressed the brake pedal, both stop lights came on but no pilot or speedo light
that’s it! I thought . . . then I noticed the front sidecar light was also on.
Next I disconnected the bike’s tail light (so both tail lights now disconnected). Result was the same as before (apart from, of course, the bike’s tail light).
So KiwiGF is right in his thinking but even with the live cables to the tail lights disconnected somehow there's enough current getting to the front sidecar light by operation of the stop lights.
I've put the 2.2W incandescent bulb back in the speedo and its now giving off its heart warming, dim, warm glow and all the horrid LEDs are happy again.