Do all the wires from the ignition switch go through the headlamp shell before continuing to their final connection?
Certainly White, Brown/Green and Blue wires do, because the 'next' switch in most circuits is on the handlebars, Brown/Green certainly originally connected to the pilot lamp in the headlamp reflector.
The only wire that might not is the Brown/Blue to the fuse and battery positive - if you can see it inside the headlamp shell, it does.
. with ignition and lights on, without turning the 'bars, grab the wiring between key switch binnacle, headlamp shell and the rest of the bike and pull it around; if that causes everything electrical to "die", likely a broken wire.
Re-reading my earlier post, I realise I could've explained another part better - because your bike does "start and run normal when the key is in the first “on” position", a PO has fiddled with original connections inside the headlamp shell. Ime, in addition to the other checks Peg and I suggested above, you should check the connections inside the headlamp shell to ensure the problem you're experiencing isn't simply a loose and/or corroded connection ...
I appreciate you might not relish the latter suggestion, especially as you posted earlier, "me not being an electrical guy"; however, because Lucas and the Co-op didn't make it difficult for themselves to comply with the US requirement, if I tell you what the PO should
have done, you can see whether he did do it ... or something else ...
As I say, originally on '79-on US-market bikes, the White wire from the handlebar kill switch was connected inside the headlamp shell to the Blue/Yellow wires bullet snap connector, so the bikes should only "start and run normal" with lights on, when the key is in the second “on” position.
However, Lucas normally designed wiring harness that only required same-coloured wires to be connected together. On non-US-market bikes, the White wire from the handlebar kill switch is instead simply connected to other White wires inside the headlamp shell (originally, there were two '4-way' (two wires in each end) snap connectors for all the White wires). That's all the PO should
have done so the bike "starts and runs normal when the key is in the first “on” position".