Singing is the Half the Battle
Pretty much every scene of Maj. Kong and his crew is underscored by some variation of the "The Battle Hymn of the Republic." Back in Civil War times, this was the battle song for the Union Army, and the hymn's lyrics include lots of fun stuff about a wrathful God destroying the wicked enemy. After the Civil War, "The Battle Hymn" became a patriotic song that all Americans could sing whenever they wanted to get excited about pummeling the enemy into nonexistence.
It's not too hard to figure out why Kubrick chose "The Battle Hymn of the Republic" to underscore Kong's scenes. We've got a crew of intrepid American soldiers daring against all odds to bring the wrath of God and the U.S.A down on the foes. The irony, of course, is that our brave soldiers have been tricked by their commanding officer, and all their bravery actually ends up destroying "the folks back home" along with the folks everywhere else.
If you've never heard this song, check out this YouTube version. Someone who definitely understood its warlike lyrics used it to score a WWII battle montage, complete with tons of bombs and guns. The twist? He used a version sung by the Soviet Red Army. Kubrick would have loved it.
Actually, We Probably Won't Meet Again
At the tail end of the film, British singer Vera Lynn is heard singing "We'll Meet Again" over a montage of nuclear explosions. This song was a big hit back in WWII days and was especially popular with British troops going off to war. It was meant to cheer up the guys, who more than likely would not in fact see the lovely lady back home ever again. The choice of the song would've reminded Dr. Strangelove's audience of WWII, and probably highlighted in people's minds how the events of WWII directly led to the Cold War.
The song might also refer to America and the Society Union, who were busy preparing mineshafts to survive in, just to come out after the nuclear winter ended and start it all up again.
Uplifting, we know.
Even if you don't know the historical context of the song, the irony's pretty clear. The song's wistful optimism is juxtaposed with the nuclear Armageddon ravaging the Earth. The whole world is being incinerated, so most people won't, in fact, see each other again. On the other hand, since a few people might survive in mineshafts (yeah, right), you could also see the song as a warning that when the human race eventually rejuvenates itself, it might very well find itself in the same nuclear pickle one day. (Yum, nuclear pickles. Are those the kind with horseradish?)