If 'radiation' is a contributory factor then area's with a relatively high background radiation level should have substantially more cases of ME than areas with relatively lower levels.
This pattern should be fairly easy to spot, but my remembrance of my O level geography re world distribution of sources of minerals leading to higher levels of background radiation is 'iffy'.
Some parts of Canada I seem to remember had significantly higher background radiation due to uranium deposits, some parts of Yorkshire and the NW in the UK also.
That's the sum total of my pre-existing knowledge on that subject, but, assuming the population data is there it should be easy for anyone to map ME cases onto such mineralogic geographical data, allowing for the fact that people move, and see if there is a correlation.
ETA - It appears that most of the southern hemisphere has relatively high background radiation, India, sections of China, the west coast of north America, and pretty much anywhere with mountains, fault lines or volcanic shield plateaus.
And, for some reason, Iran, which has a town/city that has the highest background radiation in the world.
So if the theory is correct.....