There's no one obvious source for the term, but a few things seem to have come together:

• Buttons as a symbol for technological simplicity emerged in the 19th century, during the late Industrial Revolution and what became called "The Machine Age" (the idea of a button you push is itself taken from the similarity in shape of buttons on a shirt)

• The idea of coupling that with the idea of technology destroying the world emerged by at leas the 1890s (there was a silly story about how Thomas Edison had buttons that could destroy countries)

• During World War II the idea of "push-button" warfare emerged with regards to missiles and other automated or semi-automated systems

• This became directly coupled to the idea of nuclear war by 1945, and became a staple way of talking about the new era of technological capabilities

There was never any single button that could start nuclear war; there are buttons involved in many stages of the process (just as there are in ordering a pizza), but it is not so easy and automated as the idea of "the button" would make it seem. It is also not as hard as some people think, however; these command and control systems are designed to turn intention into result very quickly.

I have written more about the idea of "the button" here, which has the specific references that I mention above.