Yes, the popularity of atomic superpowers and origin stories into comics relates almost directly to the use of the first nuclear bombs and to the ensuing Cold War--and reveal a lot about anxieties and hopes of the time.

Atomic Thunderbolt and Atoman were both introduced in 1946. Atoman is a nuclear physicist who absorbs the power of radiation by working with uranium. Atomic Thunderbolt, on the other hand, has his origin in a different sort of scientific experiment: a professor striving to create humans who are immune to the dangers of radiation! (It's comics, so predictably the experiment succeeded once but killed its inventor in the process.)

A really good look at the influence of nuclear bombs on superheroes is from the Golden Age version of The Atom, Al Pratt. His introduction is actually 1940--he chooses his superhero moniker based on his childhood nickname "Atomic Al" because he was small as an atom. He doesn't have superpowers...yet. He was marginally popular, and actually ceased to appear in All-American Heroes sometime in early 1945. Well! Unleash the knowledge of how powerful atoms can be on the world, and you can't have your Atom languish in obscurity. DC brought him back--and in 1948, they retconned in superpowers for him: super-strength, including the "atomic punch," derived from exposure to radiation during a WW2-era battle.

The early 1950s were a lull in the publication of superhero comics, but atomic awareness was still quite prominent. The Attack comic published a run of Atomic Attack! in 1953 of World War III-type stories. Michael Scheibach, Atomic Narratives and American Youth: Coming of Age with the Atom, 1945-1955, discusses how comic books were also used to serve an educational purpose, from having heroes experience the portrayed-as-mild impact of a nuclear bomb test to full-blown Very Special Episode-type stuff like Dagwood Splits the Atom (from the Blondie comic).

And from the "hey kids, it's not all bad" camp, my personal favorite is Atomic Rabbit, brave defender of Rabbitville, who gains superpowers by eating uranium-enriched carrots.

Radiation-induced mutation has been known by scientists since 1926, when it was used as a tool to study genetics (the work was done by Hermann J. Muller, working with fruit flies in the famous laboratory of T.H. Morgan at Columbia University). Prior to that, there were already radiation "fads" in popular culture, but they were about radiation making you healthy, not leading to the creation of mutant offspring. The death of Pittsburgh socialite Eben Myers from massive over exposures to radium-based "therapies" in 1931 caused the first majorly negative press about radiation in the United States.

Which does not quite give us an exact answer to your question, but does seem to line up with the emergence of some of these tropes. In the 1936 movie, The Invisible Ray, a scientist (played by Boris Karloff, no less) developed a radium-based beam that could destroy cities, but in the process gave himself radiation-related powers (glowed in the dark, could kill at a touch), though these seem more related to exposure than mutation.

From what I can tell, these sorts of "power" tropes were pretty rare, though, until the creation of nuclear weapons, in which all manner of radiation-related tropes sprung up in popular media, from the "massive bugs" to the "superpowers" approach. In the early 1960s, Stan Lee spawned a whole series of "radiation-induced powered" superheroes (Fantastic Four = 1961; Spider-Man = 1962; Incredible Hulk = 1962; Daredevil = 1964), perhaps not coincidentally linked to the major debates about nuclear fallout that were taking place (and would lead to the signing of the Limited Test Ban Treaty of 1963), much less other nuclear issues at the height of the transition from the "atomic age" to the "nuclear age" in popular lingo (civilian nuclear power = 1957; Cuban Missile Crisis = 1962). I am sure some comic buffs can find a few earlier characters, but it is interesting to me how many of these characters spring up in the 1960s (Captain Atom = 1960, Dr. Solar = 1962, Negative Man = 1963, Nukla = 1965), and how many of the original "powers" characters got their powers from other sources (Superman = extraterrestrial; Captain Marvel = magic; The Flash, Captain America = chemicals/serums). (While digging around, I also came across "Atom Man," from 1950, but notably he gets no biological powers from the radiation, he just uses radiation-powered weaponry.)

I am not a comics buff, so I am going for some low-hanging fruit above. I would be surprised if there were no radioactive-powered superheroes or villains before the 1960s, but it is interesting how the trope seems to really take off then. (Edited to add that I see /u/sunagainstgold has a few interesting pre-1960s examples.)

On the history of the imagery of radioactivity, my favorite book, which is quite wonderful, is Spencer Weart's Nuclear Fear: A History of Images (1988), and its revised edition, The Rise of Nuclear Fear (2012). This has immense and valuable information about the early (pre-World War II) pop culture aspects of radioactivity. My friend and colleague Luis Campos also has a wonderful book on the science side of this early period that came out recently: Radium and the Secret of Life (2015).