Not to nit pick but the Google says Skeeter Davis’ version came out in June of ‘62 and the Cuban Missile Crisis was in October of ‘62.
You’re right, “The End of the World” was recorded in June, 1962, so maybe it was inspired by the Berlin crisis. But it wasn’t released late in the year. It entered the country-western charts in December and the Billboard Hot Hundred chart of pop hits in January, 1963.