In 24 years and 6 months of operations, the BA and AF Concordes never had a radiation alarm caused by solar/cosmic radiation at 55-60,000 ft. 1978, 1989 and 2000 were maximum solar activity years.
They had alarms at normal altitudes when subsonic overland, caused by ground sources which presumably all aircraft on the same flighpath flew through. The last incident was in 1999, caused by a source at the former Republic Aviation factory site at Palmingdale, which affected a BA Concorde on a charter 6000 ft above.
Radiation readings are taken before and after all Concorde flights, I can tell you that normally there is hardly any difference between the two.
Many of our flight-crew have flown the aircraft for many years, some for more than 20. No unusual incidence of cancers have been detected.
I agree that radiation exposure needs further research, but airlines have to be careful. Before you know it, frequent flyers who get cancer for whatever reason, might start taking legal action against airlines, thus making our lawyer-choked societies one step closer to grinding to a halt, just to make some professional bullshitters even richer.