Earlier quoted context omitted.
Note that a great deal of those tests were underground - the Limited Test Ban Treaty banned all atmospheric tests and was signed in 1963. This was primarily because of the risk posed by fallout. Needless to say, wartime use of nuclear weapons would not be underground.
Fallout is of course dangerous. It just isn't as dangerous as it is commonly thought. We actually had a recent demonstration of this fact courtesy of Fukushima, and people in California being worried about the fallout. These worries were many, many orders of magnitude away from being well-founded.
It is instructive that the US, UK and USSR were able to agree, at the height of the Cold War (this was less than a year after the Cuban Missile Crisis), to ban atmospheric tests.