Heck, REM’s Radio Free Europe could certainly be considered a Cold War song.
Prince’s 1999.
Black Sabbath, War Pigs.
Jackson Browne’s “Lives in the Balance”. Bruce Cockburn’s “If I had a ROcket Launcher”. Just off the top of the head.
“Rocket Launcher” is actually about the Guatemala Civil War; however it arguably that was linked to cold war manipulation by superpowers. I have already added it, because I love Bruce Cockburn. I’ve seen him live about 5 times, I’d do it again in a heartbeat. Interestingly I grew up with his records in the house, my dad liked Cockburn’s music until “he became a commie pinko.”
Josh Ritter’s “The Temptation Of Adam”
Pink Floyd’s Two Suns in the Sunset. (The second sun being a nuclear explosion.)
Europe’s “The Final Countdown.”
Sent from my adequate mobile device.
“We will all go together when we go” by Tom Lehrer. 
Fishbone’s Party At Ground Zero
Party at ground zero
A “B” movie starring you
And the world will turn to flowing
Pink vapor stew
Ozzy’s Thank God For The Bomb
Then there’s Land of Confusion-Genesis
And Sting’s Russians
Fishbone Party at Ground Zero
Edit: Coke to mix
Thank you!
“Everybody’s got a bomb, we could all die any day.”
Also, Ray Gun Omics by Project Future (1984?) had lyrics such as: “Ray gun omics, more atomics! Fight inflation, with taxation!”
All She Wants To Do Is Dance is another possible contender about Cold War US involvement in poor countries with questionable regimes, isn’t it? ymmv…
Found it! 1983
“California” by Flash and the Pan.
Another Tom Lehrer: “So Long, Mom.”
There was an alternative version to Radio Clash with apocalyptic lyrics going something like:
This is radio Clash and we count your thoughts
When lightning splits the sky in half sharper than a sword
And in the thunder cloud you see another cloud
Bones as big as sticks
Even angels go to Hell on the morning of the 6…66
My attempts to find that version of Radio Clash have proven futile at this moment but it’s on Youtube.
Thanks everyone. Many of these I know and have forgotten, or never associated with war. Some are new. This is going to be quite the play list.
Iron Maiden, 2 Minutes to Midnight
Let’s move back to the 60’s: The Sun is Burning- Simon & Garfunkel
99 Luftballons and Eve of Destruction are great examples…
Not gonna look up YouTube links on my phone, but:
Russians, Sting
Land of Confusion, Phil Collins (esp. the video)
Two Tribes, Frankie Goes To Hollywood
Well, um, how about “War” by Edwin Starr?
Dancing With Tears in my Eyes - Ultravox
For No One - Barclay James Harvest
Heaven- Psychedelic Furs
I Was Only 19 - Redgum