Well, I tried out Pete's suggestion for testing except I didn't get to test it while wobbling the indicator stalk because I have found what I think is the problem before I got to that.
I found that the indicator comes on when I put it in 4th gear. Also, the left indicator is the only one that comes on when I put it in reverse. Additionally, the rubber support on the gearshift is broken (has been broken for ages but I never got around to it

).
So my suspicion now is that the switch is triggering for the reverse lights when I put it in 4th gear (which, of course is right next to reverse). This may be due to a change in alignment because of the broken support, or some other electrical problem (broken or misaligned switch, earthing problem etc).
I also need to investigate why the right rear indicator isn't coming on when I put it in reverse (possibly due to the same problem).
It also appears as though the previous issues were unrelated (I can't see how they are related anyway. The previous indicator issue caused both indicators to flash, not just the back one).
I will replace the support on the gearshift over the weekend and see if that helps, then start investigating the switch and wiring if necessary.