"Uranium Fever" and "Uranium Rock" are references to uranium mining, which became a "craze" in the United States in the mid-1950s after the US Atomic Energy Commission set an artificially high price on uranium ore and created a reward system for prospectors. This was done in order to rapidly develop domestic sources of uranium ore when the US feared that it lacked enough uranium for its rapidly-expanding nuclear arsenal and feared its over-dependence on foreign sources (like the mines in the Congo). There are all manner of uranium-mining kitsch from those days, like the game "Uranium Rush" from 1955 (which I am happy to say I own a copy of — my wife did her dissertation work on the history of uranium mining in the US Southwest). There were a few spectacular stories of prospectors striking it rich (e.g. Vernon Pick and Charlie Steen) but most uranium miners did not fare too well (and in the long term, their exposure to radioactive radon gas increased their chances of lung cancer dramatically; some were later compensated under the Radiation Exposure Compensation Act).

So I would put these in a very separate category than the ones that laugh about the bomb. There were songs that laughed about the bomb, in a dark humor. There were also songs that very earnestly fretted about the bomb, like Fred Kirby's "When that Hell Bomb Falls" (1950), Al Rogers' "Hydrogen Bomb" (1954), or the Talbot Brother's "Atomic Nightmare" (1957). The lyrics of these songs are pretty dark, even if the melodies are not. I don't think they're meant to be that humorous.

Most of these songs are what we would think of as novelty songs, and were not big hits in any respect, though some were made by pretty big artists (e.g. Bill Haley and the Comet's "Thirteen Women" (1954)).

I think it would be wrong to conclude that these people (or the American people) did not understand the seriousness of the technology — they understood it in a way that most people do not understand it today, and felt it to be a real threat. But humor is one way in which a society works through its anxieties. And in any case, it was a major issue of the newspapers of the day — so it's not surprising that it ended up in other forms of popular culture, like music.

There is a really amazing set of these songs published under the name "Atomic Platters", and with an excellent compendium book that talks about the various songs and their lyrics. The themes change over time, reflective of different general shifts in discourse about the bomb (e.g. the very early songs are a bit playful, by the 1950s they have undertones of anxiety, and in the 1960s it moves on to Strangelovian farce).

The really interesting shift, in my opinion, is what you start to get in the early 1980s, during what is called "Cold War II" by some historians, and the "War Scare" by others. If you were going to chart American and European anxieties about the bomb, they start out low (but palpable) during the period of American monopoly (1945-1949), spike in the early 1950s (Soviets have the bomb, H-bomb developed), peak in the early 1960s (Cuban Missile Crisis), then decrease palpably over the 1970s (detente), then peak again in the early 1980s (Reagan), then start a downward slide that dips dramatically in the early post Cold War and then maybe pops up again in the late 1990s and after 9/11. The songs of the early 1980s make an interesting contrast to those of the 1950s — very different sensibility about music, about nuclear technology, about how these issues can be addressed, e.g. Nena's "99 Red Balloons" (1983), which is very explicitly about the dangers of accidental nuclear war (which I recently wrote a bit about here).

I think it's easy to overestimate the importance of some of these songs in their day, in any case. Most of these were not hits, were one-off B-sides put out by artists trying to capitalize on song themes that were "ripped from the headlines." If you select all of such songs (as the Fallout series does, as does the Atomic Platters album), it can look like a lot of attention to this theme, but they are a drop in the bucket compared to the total music output of these times.

While looking for a particular report about Atoms for Peace, I found something in the National Security Archive at George Washington University website that might be of interest.

It's a report from the Energy Conservation and Power subcommittee titled American Nuclear Guenea Pigs : Three Decades of Radiation Experiments on U.S. Citizens and it includes brief descriptions of a number of experiments involving exposing people to radiation both medically and otherwise and has this rather scathing opening paragraph

Documents provided by the Department of Energy reveal the frequent and systematic use of human subjects as guinea pigs for radiation experiments. Some experiments were conducted in the 1940s at the dawn of the nuclear age, and might be attributed to an ignorance of the long term effects of radiation exposure or to the atomic hubris that accompanied the making of the first nuclear bombs. But other experiments were conducted during the supposedly more enlightened 1960s and 1970s. In either event such experiments cannot be excused.

You may have also seen this footage from nuclear tests that features among other things, soldiers advancing towards a mushroom cloud from a nuclear detonation apparently unconcerned and without obvious protective measures.

There's also a book titled The Rise of Nuclear Fear by Spencer R. Weart that I've been lead to believe would actually probably answer your question and more, but I have yet to actually open the copy I ordered on a friends recommendation.

With all of that in mind, I would argue that the average person in the early 1950s would have a limited understanding of the dangers beyond it being generally dangerous, especially given instances like the infamous John Wayne film The Conquerer filming downwind of nuclear test sites.

Dr. Strangelove is right at the heart of this question. Kubrick's 1964 black comedy was the first major work to use satire, including music, to disarm the subject of nuclear war.

I know this topic is about the 50s, but I wanted to add that this is not the only case of approaching nuclear technology (and disasters) in a for our understanding in a seemingly grotesque way.

I'm writing this because I remember a certain song, 'Burli' form a austrian pop band (Erste Allgemeine Verunsicherung).

This song came out only one year after the disaster in Chernobyl and is about a (finctional, of course) child born shortly after said disaster, who develops numerous mutations. They sing about his life and how he later falls in love with a girl from the neighbourhood having a similar case of mutations caused by radioactivity.

The song is very light-hearted and funny (at least they inteded to) but considering the catastrophe of Chernobyl this seems shocking. Of course one could interpret the song as satire, but considering the history of the band, which mostly includes purely comedic songs this is certainly debateable.

This might be more than you're asking for, but for a very full discussion of the cultural changes in the early atomic era, you might want to check out Paul Boyer's By The Bomb's Early Light. It talks about popular culture, politics, and religion, among other things.