So which origin for Godzilla is more common at this point, that he was something modern that was mutated by the nuclear bombs, or something ancient that was awakened by him? It seems like the different versions have switched back and forth between the two origins a lot over the years.
Showa continuity: The three known members of the Godzilla species (the one killed in '54, the one featured from '55 onward, and Minilla) were naturally giant prehistoric dinosaurs that lived in the ocean depths, and were turned radioactive and displaced from their natural feeding grounds by the Marshall Islands nuclear tests. (At least, that's what the original film established, though
King Kong vs. Godzilla implied that Godzilla had been "created" by humans without going into specifics.) They weren't "awakened," but were just a surviving, undiscovered fossil species like the coelacanth.
Heisei continuity: Godzilla was a carnosaur of a species that had survived to modern times on a remote island, but was mutated to giant size by exposure to radiation. Through the convoluted logic of
Godzilla vs. King Ghidorah's time-loop rigmarole, the same thing happened to create the second Godzilla that emerged in 1984 (which the time travelers mistook for the one killed in '54). Similarly, Baby Godzilla (aka Little Godzilla aka Godzilla Junior) started out as a human-sized infant but was turned giant between movies by radiation exposure.
Millennium continuities: Unspecified, but they're all sequels to the original film that's shared by the Showa and Heisei continuities, and thus Godzilla can be presumed to have one of those two origins.
TriStar: The "Godzillas" of the 1998 film and animated series were a newly emerged, radiation-induced mutant species descended from iguanas.
Legendary continuity: Godzilla is a naturally evolved ancient Titan drawn to the surface by the rise in radiation levels after 1945.
Shin Godzilla: A new mutation arising from
mixotrophic sea life mutated by radioactive waste and given the ability to auto-evolve into new forms.
Godzilla Earth anime trilogy: Believed to be the end result of natural selection, newly evolved and gaining useful genetic traits via horizontal gene transfer from other life forms, but possibly predestined to arise in order to punish human civilization for despoiling the Earth.
Singular Point: Kaiju including Godzilla are the result of Earth organisms being mutated by an extrauniversal substance called Red Dust, comprising multidimensional Singular Points (singularities) where normal physical law breaks down. However, the first known Godzilla emerged and was somehow killed c. 1950, with a second one emerging in 2030.
So it hasn't really alternated between the two. The first 2-3 eras portrayed Godzillas as a surviving fossil species that was altered one way or another by radiation, while the Japanese versions of the past decade have consistently gone with the "newly evolved organism" origin, and Legendary has gone with the "reawakened ancient creature" origin that in the Showa era was used for kaiju like Rodan and Gamera (and the Rhedosaur in the movie that started it all,
The Beast from 20,000 Fathoms) rather than Godzilla.