According to the Corpus of Historical American English (COHA) at BYU, snafu began appearing in popular print in the 1940s; in fact, the 1940s is the decade which has the most citations, at 22. For example, an "Army & Navy: Joe & Joe" column in the November 30, 1942 issue of Time provides the basic definition and a few examples.
This matches the Google Ngram. Appearances prior to 1940 appear to be OCR errors or nonsense words, but by 1943 you can find the term used in publications ranging from Yachting magazine to Life to the University of Michigan alumni magazine.
See Google NGram
A letter to the editor in the same complains:
… [Y]ou use the word "snafu." I find this puzzling because it is not listed in any dictionary, nor does it sound like any slang I have heard before.
to which the editors respond
Snafu is a service slang word compounded out of the initial letters of the words in a phrase politely translated: "Situation normal, all fouled up." Pernicious snafu is somethings called susfu, or "Situation unchanged,still fouled up."
By the end of the war, evidently, editors no longer needed to define the term for their audiences. Either its use in plays, films, and publications of the era or the omnipresence of the phenomenon in wartime had made it singularly useful.