Not quite. In the original film, Godzilla was not created by the bombs; he was a member of a species that had survived since prehistoric times in the ocean depths, much like the coelacanth.
The people of Otoshima had worshipped and sacrificed to him as a god for generations. The Marshall Islands nuclear tests displaced him from his normal feeding grounds, driving him to seek new food sources in Japan. Apparently they also made him radioactive, hence his atomic breath. And this same thing happened to three known members of the species: the original Godzilla killed in the first film, the second Godzilla featured from Godzilla Raids Again through Terror of Mechagodzilla, and Minilla, the juvenile member of the species. (Something very similar also happened to the giant octopus in Ray Harryhausen's It Came from Beneath the Sea, a year after Gojira. Those selfsame Marshall Islands tests contaminated a naturally occurring giant octopus, and the fish it preyed upon could sense its radioactivity and were able to flee its approach, so it had to seek out new food sources that couldn't sense radioactivity, such as humans.)
The idea of Godzilla as a mutant created by atomic radiation didn't come along until Godzilla vs. King Ghidorah in 1991.
besides those running people of course...
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