During which two decades of the 1900's was the threat of nuclear war the greatest?

1950s and 1960s.

Explanation:

The 1950s and 1960s were the two decades with the greatest probability of a nuclear war. The first decade was characterized by the development of nuclear warfare by the United States and the Soviet Union such as fission bombs, Hydrogen bombs and fussion bombs, as well as prototypes such as nuclear-propelled bomber. Lastly, the second decade was marked by the event of Missile Crisis in Cuba, where Soviets installed missiles on request of Communist regime ruled by Fidel Castro.