According to this visualization there were no nuclear detonations in 1959, and only 3 by France in 1960. Why were there no nuclear detonations by the USA or USSR in 1959 and 1960 but over 50 in each years before and after this period?
Why were there no nuclear detonations in 1959?
The reason there were no tests in 1959 was that the Soviet Union, Great Britain, and the United States agreed to a moratorium on nuclear weapon tests in 1958. This moratorium lasted from November 1958 to August 1961. The Soviet Union resumed on 1 September, 1961, with the US following suit a couple of weeks later.
France, being on the verge of being a nuclear-capable nation in 1958, did not take part in that moratorium. They didn't test in 1959 because they did not quite have the ability to do so. They did have that ability in 1960.