Just a bit of an update for anyone interested.
Removed the sidecar mudguard with its cabling, brought it in the warm house to redo the wiring and simplify it by splicing the live and earth cables (see my updated sidecar diagram) with Posi-Lock and Posi-Tap connectors and changed, the now unnecessary, 5 pin Superseal for a 3 pin.
Tested it all for continuity with my multi-meter and everything ok, removed the battery from my bike and brought that into the house and with test leads checked that all the lights (LEDs) worked as they should, everything ok.
Back out to the cold garage to check my bike's wiring. Apart from the Quartz Halogen headlight my A10 has all LED bulbs, including pilot and speedo bulbs, and I went through it systematically checking all the relevant wiring and connectors on the bike. I also checked the correct filament was coming on for the stop light and replaced the 3 pin Superseal connector's other half on the bike. So all the relevant wiring had been checked or redone.
Turned on the light switch - all ok. Pressed down the brake pedal - all ok.
Went into the house for the mudguard and joined the Superseal connector. Carried out the same test as I did for the bike but this time (and as previously) when the brake pedal was depressed the pilot light, speedo light and front sidecar light came on.
Ok,
change all the LEDs on bike and sidecar back to the normal incandescent bulbs and retest. The result - everything working fine.
Changed the bulbs again for LEDs - same problem. Swapped around the sidecar's and bike's stop light LEDs - no difference.
Removed the sidecar's front LED and replaced it with a 10W incandescent bulb. Now everything works as it should.
Now what! I'll tell you - I'm going to leave it at that, with a 10W incandescent bulb in the sidecar's front light, as it hardly gets switched on anyway.
However, if anyone can see anything wrong I'll be very happy.