Before 1940, the word appears in metallugical literature, and occasionally in discussions of the manufacture of ice cream. Its frequency is approximately 0.0000001000%, as the Ngram Viewer likes to say, or 1 in 1 billion.*
Today we know "meltdown" best as a word for an accident in an overheated nuclear reactor.
"Meltdown" starts to appear in nuclear literature to describe an accident with the EBR-1 breeder reactor at Arco, Idaho, in 1955. For an example, see "How Safe Are Our A-Power Plants?" from Popular Science for November 1956. I presume the nuclear engineers got it from the discourse of metallurgy and materials science, which play a vital role in reactor design. Ice cream-- not so much.
By 1960 there are many appearances of "meltdown" in nuclear engineering literature. It has climbed to around 2 words in 100 million. Between 1962 and 1965, almost half the appearances of "meltdown" in the Google Books English corpus are accompanied by the word "reactor." A search for "meltdown" without the words "nuclear," "reactor," or "atomic" reveals that the metallurgists and dairymen continue to use it.
The event that made "meltdown" a household word was the accident at the Three Mile Island reactor in Pennsylvania in 1979, just a few weeks after the release of the film The China Syndrome. (I have not determined whether the word "meltdown" is spoken in the movie.)
Rapidly "meltdown" shoots up to around 1.5 words in 10 million. At first, the growth is due to the rise of nuclear safety as a topic of wider general discussion. Eventually, writers seize upon it as a metaphor. There are mental meltdowns and financial meltdowns. In the late Nineties, it takes another jump, rising to about 4.5 words in 10 million, where it remains today. (Well, not today today. Until things cool down in Japan, we will be saying "meltdown" to each other a bit more often than we usually do.)
Google searches of the Web (rather than Books) reports 1.3 million hits on "financial meltdown" and 124,000 hits on "emotional meltdown." The Google News Archive suggests that in 2009, journalists reported that the Labour Party, Serena Williams, Wall Street, and the Copenhagen global warming talks all experienced meltdowns.
*Dear Google: I would welcome an option to label the axes of Ngram Viewer charts in scientific notation.