Update. My Mini is running!
A friend who is very good at fixing mechanical things came round today to help me fix the Mini.
To recap, since it died I have replaced the points, condensor, rotor arm, dizzy cap, spark plugs and HT leads. Plus the vacuum advance hose, because it's rubber bushings were looking quite tired. (Also a new air filter, oil filter and oil change, prior to this)
We started by checking voltage to the coil. 13v present and correct. Then he directly attached a plug to the coil (as per sonscar's suggestion above). It had no spark at the coil, but then after some tinkering we got a good fat spark. To be honest I have no idea what that tinkering was. Anyway, with the cap off and a nice spark at the points, the Dizzy cap went back on and we had good sparks at all the plugs. But it still wouldn't start. Plugs out, lots of compression, so that was alright.
My friend thought the plug leads were in the wrong places, and had got them in what appeared to be the right order. We looked again at this, and suspected we were 180 degrees out, so we swapped them over.
That got it running on two cylinders. Then we swapped two around again, and we got all 4 cylinders back again!
The engine ran but was running very fast and the revs weren't dropping, so he gave the carb a good clean and check over, adjusted the throttle cable, and got it idling properly.
We suspect the coil is breaking down and caused all this (although the old points were looking quite fried). A new coil will be ordered today.
And an electronic ignition conversion while I am at it, now that it is running properly again.
(Would you still call it modernising when you would be fitting something to a 1991 car that even my old Vespa had back in 1978?)
I am going to give it a proper test drive now, and see if it has any more issues.