Cities in Dust - Siouxsie & The Banshees
London Calling - The Clash
Five Minutes - Bonzo Goes to Washington
Bri2k
This Is Not America - David Bowie
There’s Bragg’s The Marching Song of the Covert Battalions, at least the first verse.
And New Model Army songs such as 51st State, while not directly about the Cold War, certainly are about the reaction to American Imperialism that was part-and-parcel with it - the whole CND-Greenham Common-Thatcherite Britain vibe
I don’t think Every Day isLike Sunday counts as a “protest” song. In that one, Morrissey’s actually pining to be burnt to a crisp a world full of such misery that one’s underwear gets stolen.
So Afraid of the Russiansby Made for TV
Good point. That morose bastard actually wants to be nuked. 
Americans-Soviets by CCCP
There’s really nothing conclusive there. The only thing Pat MacDonald actually said was that many people falsely think the song is positive when it’s supposed to be grim. All that reveals is that a lot of people are incredibly oblivious to irony. The part about the song being about nuclear war is just “hinted at”.
I think the song is about individuals not nations. It’s talking about how people ignore the moral and social impact of their actions in pursuit of personal gain (using sunglasses as a metaphor for this willful blindness). MacDonald mentions a nuclear scientist but he could just as easily have been talking about a genetics researcher or a corporate executive.
Speaking of XTC - “Generals and Majors”
It goes from generic anti-War song (like Sabbath’s ‘War Pigs’) to Cold War with:
“Your World War III is drawing near”
Also, don’t forget the Dead Kennedy’s “We Got a Bigger Problem Now”*, with such fine lines as:
Welcome to 1984
Are you ready for the Third World War?!?
You too will meet the secret police
They’ll draft you and they’ll jail your niece
You’ll go quietly to boot camp
They’ll shoot you dead, make you a man
Later lines reference El Salvador and Afghanistan, so Cold War it is.
*A response I think to their earlier song “California Uber Alles”, about Jerry Brown becoming President - the bigger problem being Ronald Reagan, of course.
Just thought about “Afraid of the Russians” by Made For TV, and stumbled across this site - have fun!
Fishbone - Party at Ground Zero
And if you want to go the other way, there’s the ludicrously jingoist VOA by Sammy Hagar. The video is cheesetastically awesome. Hagar and Reagan kicking commie ass!!
Not technically a protest song, but cold war/nuclear armageddon themed:
Ultravox - “Dancing With Tears in my Eyes”