Just to clarify: By radiation sickness, I mean this: https://en.wikipedia.org/wiki/Acute_radiation_syndromeWell, I would argue that generational birth defects bring radiation sickness to those not present.
Early nuke designs just weren't designed to emit large doses over a extended time - they're designed to act more like a large scale fire-bombing, just packed into a single bomb. Nukes making their impact area inaccesible for years to come are intentional designs of later decades.
All the other radiation-induced diseases hit way slower and more subtile - and it's a statistical conclusion, rather than a personal one. Either way, these things don't kill entire elite divisions (consisting of fit and able-bodied men in their best years) in a matter of days.