When we talk about the Cold War, you have to remember that there were entirely different decades with different attitudes and anxieties towards the threat of a nuclear exchange. Allow me to elaborate as this is actually very interesting stuff, starting with the 1950s as OP brought up:

##1950s

This was shortly after World War II, and when the Soviets made their own H-bomb. Attitudes in the US were more jingoistic and pro-nuke than you'd expect. People didn't really understand what an all-out nuclear war implied. They imagined it more like the Soviets would bomb some cities, but they'd still be able to rebuild like Europe did after WWII. A lot of people didn't understand what fallout was.

This is when propaganda videos like, "Duck and Cover" came out. We look at it today and think it sounds ridiculous, but people back then really believed that was what could be expected.

Don't get me wrong, there were still some people who were more leery of World War III, but most of them didn't really grasp just how bad it would be. Humor is a common technique for channeling anxiety. That's when we got songs that were cheerful, but also kind of scared of the potential for this new destructive weapon. The somewhat dark lyrics contrasted against the upbeat melody.

##1960s

This is when people started to get more concerned. The Cuban Missile Crisis made people genuinely afraid of a nuclear war scenario with the USSR. There were some people who advocated for nuclear disarmament. Songs reflected these fears, lyrics became more overtly scared and dreary. Songs like, "We'll all go together when we go", for example.

##1970s

People began to breathe a sigh of relief, but tensions across the Iron Curtain were still definitely there. Tensions began to ease, direct communications between both sides were established. Treaties were signed, people in the US began to relax a little. This is what is known as the Detente. Of course, this didn't last for too long. I don't know too much about songs relating to nuclear war in this era, but fears definitely started to die down a bit.

##1980s

Now we get back to heightened levels of fear among the American population. Next to the Cuban Missile Crisis, this would probably be the closest the world got to a nuclear war. Reagan came into power and said the Soviets had used the Detente to achieve their aims. Ronald Reagan's language was much more aggressive. The Soviets were concerned he was planning a preemptive strike. This is also when the Strategic Defense Initiative was started, worsening tensions even more. Both sides had their fingers practically on the trigger. There were multiple, "False alarms" that could easily have dragged us into nuclear war.

Music also portrayed these renewed anxieties with songs like, "99 Red Balloons" that feared an accidental trigger of WWIII.