In the 1920s-1940s there was considerable scientific work on the effects of ionizing radiation to induce mutations, originally come out of work by H.J. Muller on fruit flies (for which he won the Nobel Prize in Medicine for 1946), but later expanded to all sorts of agricultural breeding programs. It isn't, of course, that radiation makes things necessarily bigger or anything like that, but it can increase the number of mutations far above the normal rate. If you irradiate thousands of fruit fly eggs, you'll end up with a few very bizarre specimens as a result (e.g., flies born without eyes, flies born with weird wings, whatever).

Anyway, my sense is that this work got popularized after the development of the atomic bomb — again, Muller's Nobel came at a pretty opportune time for that — and concerns about radiation and mutation were very high in the immediate postwar period. Obviously the main medical concerns were cancer and birth defects — nothing positive.

The other potential cultural thread here is that there had been associations between radioactivity and health since the 1910s, and several decades of radioactive health fads (radioactive water, etc.). These eventually slipped into horror when it became clear that this could be quite dangerous, both with the famous Radium Girls case of the 1930s, and also with the sensational case of Eben Byers, a millionaire who drank radioactive water until his jaw fell off and he died, in 1932. Which is to say, this is part of the zeitgeist of radiation as well.

There may have been radioactivity and monster associations through the 1940s but I think of the canonical entries into that science fiction genre as coming in the 1950s: Gojira/Godzilla and Them! in 1954. Godzilla is about how a nuclear explosion awakens and possibly supercharges some ancient sea monster, an elaborate metaphor about nuclear testing and the Japanese experience of the atomic bombs (and the Castle Bravo accident), and is less obviously directly related to super powers and mutated bigness (at least in his original incarnation, the H-bomb seems to has mostly just "woken him up" as opposed to "made him"). Them! is an American film about nuclear testing creating giant ants in Nevada. The latter is easier to see as exactly representative of the genre you are asking about, and is a combination of anxieties about the domestic testing of nuclear weapons and the effects of radioactivity.

Jumping from these things to "super powered heroes" takes a leap that I don't quite know the origins of, and perhaps someone else can write on that. The main "radioactive heroes" that come to my mind — Spider-Man and the Incredible Hulk — both date to 1962, which is probably not coincidentally also a year in which Americans had a lot of anxiety about nuclear testing and fallout, and is right before the creation of the Limited Test Ban Treaty that banished atmospheric nuclear testing, largely out of health concerns. However the genre might be a lot older than that, I don't know, but it is, again, likely a result of the various scientific and cultural forces previously mentioned.

The best overall history of these kinds of tropes is Spencer Weart's Nuclear Fear: A History of Images. Luis Campos' Radium and the Secret of Life goes into the early scientific work (like Muller) and its relationship to the early tropes about radiation's pros and cons.