That's the thing: Dr. Yamane's wife/Emiko's mom is not seen or even mentioned in either version of the film. What more, this film doesn't take place in an alternate universe where Hiroshima and Nagasaki are not devastated by atomic bombs. It happened and its heavily implied (and several time, almost outright stated) to be on everyone's minds. So the bombings happened, Dr. Yamane's wife is not present and...
Well, there are several reasons for Mrs. Yamane to be in either Hiroshima or Nagasaki without Dr. Yamane or Emiko (friends, family). Dr. Yamane would not want any other family to go through what he and Emiko went through, hence almost complete investment in Godzilla's well-being... at least until Godzilla trashes Tokyo and Dr. Yamane sees that Godzilla is just going to keep doing what Fat Man and Little Boy did.