"West of the Wall" - far too old for most of you to remember.
Phil Spector type background, girl singer with pain in her voice
West of the Wall, I’ll wait for you
West of the Wall, my heart will still be true
Berlin, in case anyone needed to know.
Nice beat, and you could dance to it.
Since nobody’s mentioned it, how ‘bout the "Feel Like I’m Fixin’ to Die Rag" by Country Joe and the Fish? (“Be the first one on your block to have you boy come home in a box!”)
“Political Science” by Randy Newman
“Two Tribes” by Frankie Goes to Hollywood
“Eve of Destruction” by Barry McGuire
“Masters of War” by Bob Dylan
Keith
My apologies – Billy Joel’s song about a circus clown is “Leningrad”, not “Stalingrad”.
Also, Doug Bowe, thank you for the Kingston Trio one! I had a professor who quoted that song once but didn’t know the name of it until now – in fact I was just about to post the question this morning (after spending way too much time looking in Kingston Trio lyrics archives).
A few other songs –
“Crazy Train” has some lyrics – “millions of people living as foes”, “heirs of a Cold War, that’s what we’ve become”
“Forever Young” by Alphaville – “Are you going to drop the bomb or not?”
Tom Lehrer had a bunch of good songs, as noted. One more concerned a plan for Western nations to join in a multilateral force (MLF) to create a nuclear deterrent to Communist aggression - resulting in the “MLF Lullaby”:
*"Why shouldn’t they (the Germans) have nuclear warheads?
England says no, but they all are soreheads.
I say a bygone should be a bygone
We’ll make peace, the way we did in Stanleyville and Saigon.
Some say the Germans are evil and mean
But that won’t happen again
We taught them a lesson in 1918
And they’ve hardly bothered us since then."*
And for the most unintentionally hilarious Vietnam War song, there’s the Battle Hymn of Lieutenant Calley.
New Frontier- Donald Fagen (In which our narrator takes his date into his parents’ bomb shelter to pitch some woo.)
I like how the terminology bled into even the most simple tunes:
You dropped The Bomb on me, baby
You dropped The Bomb on me
(wheeeeeeeeeeeew)
And band names are everywhere, like the B-52s. Did they ever do a Cold War-themed song? I can’t remember any.
And then there’s U2, 7 Seconds, Nuclear Assault, Megadeth, Berlin, the Communards…
Sofa King wrote:
Yeah, and who can forget The Neutron Dance?
If memory serves, “B-52” was the name of the beehive hairstyle Cindy and/or Kate wore. Only an indirect Cold War reference.
To be precise, the Trio just sang the song. Sheldon Harnick is the song’s composer. He also did the lyrics for “Fiddler on the Roof” and wrote many other songs.
I forgot another Dutch classic, DE BOM (the bomb), from the early eighties. The lyrics translate roughly as
Making a career - until the bomb drops
Working on my future - until the bomb drops
I rush through my diary - until the bomb drops
Safe in my insurance scheme - until the bomb drops
And when the bomb drops…
The I lie in my best suit
With my classifications and my cheques
My insurance and my knowledge of words
Under the ruins of the city next to you
etcetera…
Doesn’t Coldfire chip in on these subjects?
Pretty much the whole of Roger Waters’ Radio KAOS
The tide is turning… 
Though it doesn’t mention the USSR, “The Smallest Astronaut” by The Royal Guardsmen.
“Surfin U.S.S.R.” by Ray Stevens.
Wasn’t “Major Tom” a cold war hit? I still like it even if it doesn’t count for you.
The cold war includes Vietnam also.
“Jimmy Newman” by Tom Paxton. “Whose Garden Was This” also by Tom Paxton.
How about “Mother Russia” by Renaissance?
Annie Haslam had a terrific voice.