Electrical update:
Everything seems back to normal. The original bullet Y-connections behind the grille where the headlight and parking light circuits branch were dubious and the branch across to the left hand lights was about 6" or 8" too short and were running diagonally across in front of the alternator, supported with electrical tape to the upright strut. Not pretty, in the way and prone to dirt and water. I extended the ends from the main harness in 2 directions (soldered Y-junctions) and added 4-circuit line plugs each end to connect the headlight branched. It all now tucks up into the slam panel arch. All affected parts of the main loom and the left headlight wires received fresh wrap.

One curious discovery: The Mini has the European style headlamps with the built-in parking lights. For some strange reason the left light was fed separately from the right light by a red/black wire outside the main loom but within a loose black vinyl sheath. This seemed to be part of the original construction. The right side was fed as expected by a red/green wire coming out of the loom. With the light switch off there was continuity between the red/green and red/black so they joined somewhere, possibly at the switch. Not knowing the load on these wires, I opted to keep them both in service. 

Howard, what you need is a older brother. Or maybe it's a southern term. As a young kid my brother and cousin ( both 6 years older than me ) were all about the hard top convertibles. It didn't sound right to me but I was just a kid. The only cousin I had who's parents could afford to buy them a new car was not allowed a convertible but got the 58 or was it 59 Ford truly hard top convertible as soon as it came out. Of course that uncle was partners in the Ford dealership. Steve (CTR) 

The only terminology I've ever been familiar with are 2 door & 4 door hardtop (no B-pillar) 2 door & 4 door sedan (w/ B-pillar) convertible, and the hardtop convertibles that Ford produced in the late 50's.

Good luck sorting your electrical issue, its sounds like you're getting some good advice.

It may be a difference in terminology - back then we had sedans 2 or 4-door and 2-door convertibles. The convertibles obviously had no B- or C-pillars. hen the designers came up with a hard top style that used the doors of a convertible along with the front and rear side windows of the convertible to eliminate the B-pillar. The fixed steel roof spanned from the A- to the C-pillars. When driven with the windows down, they had the look of a convertible with the top up. Thus they were referred to as 'hard-top convertibles'. They sold well to people who wanted the style but not the short-comings of a convertible top, especially in areas that get winter. Generally, removable or folding hard tops probably came later (don't quote me on the timeline!). If I ever see the car again, I'll get more info on what it actually is and hopefully a photo. All I can say was it was huge, and seeing the Mini across that long wide hood made it look like a shopping cart.

Malsal: the fuse box is OK (tested) - it was replaced a few years back, with no foam firewall sound insulation behind it. The car is stored indoors and not winter-driven.
Thanks for the reminder.

Check the back of the fuse box for corrosion also.

Dan ... After you get your electrical issue resolved, please post a photo of a 1971 Buick hard-top convertible. From what I recall, Buick never made one. Hardtops yes. Convertibles yes again. Help me and those "young-uns" out on this one. ,

I agree with Alex.  Since the headlights did not work and now they do, that suggests either a major problem with the supply or earth cables for the whole car.  Clean all of them both in the boot and up front.

My car does indeed have the rear "fog-guard' switch but no light on the back of the car. (I recall s pare wire in the boot.. have to check it out.) According to Haynes, it is fed as far as an in-line fuse by a blue wire branching from the headlight circuit after the light switch. From the fuse a Blue/Orange feeds the fog switch.

On the Haynes chart, the hazards are fed from the in-line fuse (with the loose brown wire) by a line labeled "PU/O" at one end and "PU/W" at the hazard flasher. From there it becomes a LG/P to the hazard switch.

The large spade terminal on the solenoid is furred up - clean the ouside of the male and the inside of the female back to bright brass.

Fault finding is a nightmare as you'll have 12v everywhere until you put load on and it drops off.

Thanks for the Lucas Colours info. Great stuff.

When finding out what did not work, everything eventually worked as it should,except for the 4-way hazard lights. According to the Haynes wiring diagram this car should have separate flasher units for the turns signals and hazard signals.
(Part of the checking process was figuring out which systems required the key to be "ON", and the heat didn't help the mental process.)

You may be onto something with the idea of the powered antenna: when I got the car, it had a manky antenna (can't remember if it was powered, don't think so, but it went into the bin.) It also had a stereo unit I could never get to work properly and would not remember the station I had set it to, so it got binned too. The green/white wire I see may actually be Light Green/White, which is the same code as the original radio feed according to the Haynes chart.

I don't have my manual with me, but check if those two go to an in line fuse. One is for the hazzard lights, the other one to the rear fog light.

swap out the condensor.

When you were finding what did NOT work... did you find anything that DID work?

The headlights should not involve the ignition switch so their failure is either in the main battery feed or main earth connection.  (Relates to my question above).

For the Lucas color codes see the link below.  Green/White is for the right-side turn signals.  Purple/orange?  That's an odd one.  The link below says it's for a power antenna.  

https://www.mgexp.com/article/lucas-colours.html

It was a beautiful Sunday morning. The brakes were fixed. The carpet installation was complete. The garage was somewhat tidied up. FINALLY I could go for a short Mini drive, first one of the year, just to clear the pipes out.

She started properly. I backed out of the garage and tested the brakes in reverse. All good. No pulls. I back out onto the road and then do a forward brake test - still all good. I traverse the neighbourhood doing short accelerations and braking... just to be sure. Onto a secondary road now... past the school zone and up to 80km/h. All good. Stop for a traffic light, then make a left turn. Accelleration good. Slow for a red light at the off-ramp of the 416 highway.... oh-oh a stumble, then a stall. Able to restart and get over 2 lanes to the right shoulder where it dies.

It's hot there, no shade and I'm wondering what went wrong. The engine cranks fine but no a whisper of a fart from fumes. I look down at the gas gauge - have to squint to see that the needle is on "E". Whoah? Out of gas? But I didn't hear the electric fuel pump trying to find something to push.

Just then, a huge 1971 Buick hard-top convertible (for you young-uns, that's a 2-door car with a fixed roof and no B-pillars. A really nice car guy named Adam and his son out for a cruise. Their car has a a huge engine, which is now a little bigger than a 454 cu. in. It has the lumpiest idle I've heard since I was a young-un, and seems to smooth out probably around 1500 to 2000 rpm, I'm guessing. Adam tells me the exhaust pipes are 3" diameter - you could roll a tennis ball from one end to the other. They drive me home and back again to get a can of gas, then wait until I try to start it. No dice. Seems to be spark and fuel to the carb. Check fuses... seem to be OK but no go. They drop me off at home again where I wait for my darling wife to take me back and drag the Mini and me home.

Most of today was trying to find the gremlins. All kinds of things seemed to be not working, turn signals, 4-ways, headlights and so on. Weird. I photocopied the relevant Haynes wiring diagram, assemble an assortment of coloured pens and a highlighter and proceed to systematically trace power. I found one 3A fuse that had blown - I think it protects the fuel pump power feed which I added years ago. Behind the carb, there is an in-line fuse holder fed by a 16 or 18ga. brown wire and protecting a purple/orange that disappears back into the loom and gloom. The fuse was OK, but I later notice the brown wire had fallen out of the fuse holder... I noticed it from the sparks it made when it touched the carb. Yikes!

The headlights were not working before lunch and I accidentally left the switch on when i went for lunch. When I came back they were on.  As I traced and tested each system they seemed to come back to life. The engine even started and warmed up normally. Only the 4-ways and the interior light were still not working when I packed it in at 4 pm. (I don't know if the interior light ever worked! I guess I better go check it is still off!

A curious discovery: there are two wires ending behind the master cylinders (left-hand car, mid 80's export to Germany). That seem to serve no purpose. One is Green/white and the other is Purple/Orange (always live 12V?). They are not for the brake low-fluid warning system - that is also working properly. Any ideas?