An update for anyone who reads this:
I solved the issue, when I had a moment to properly run diagnostics.
TL:DR - The fix was to replace the PTFE linings of the hot ends.
How this pattern of error works:
I believe that the original PTFE linings were of poor quality, and began to degrade when I attempted to print in ABS at only 235-240 degrees (As reported by the Mightyboard/thermocouple combination, which tends to report around 10 degrees higher than RepRap RAMPS/Thermistor combo for the same temperature).
It’s possible that either the PTFE material was poor to begin with, or perhaps more likely the length of tube was too short, allowing a bulge to develop at the bottom, which required a higher temperature to soften it enough to work.
Then, turning up the temperature initially unblocked printing, but above 250 degrees or so it would have begun to degrade the PTFE tube significantly, so eventually the tube was ruined (fused to the also cooked filament inside).
It’s odd that the symptoms were so linear - a steadily increasing temperature at which printing would occur - and so similar for both nozzles. This didn’t help diagnostics, as it pointed me towards the thermocouples.
I have now ordered new nozzles, a short section of PTFE tube with outer diameter (OD) 3mm, inner diameter (ID) 2mm. These were cut to about 31mm long which was slightly too long, I shaved tiny amounts off until it fit up the heatbreak. It was a very tight fit to get the PTFE up the heatbreak - I first inserted some filament to guide the PTFE and keep it from compressing too much, then used needle nosed pliers to push the tube up.
Now, I have printed several items in PLA successfully, I will try ABS again and see how it goes!