What wouldve happened if the atomic bomb dropped on Hiroshima didn work?
If you assumed the same happened with Nagasaki, and the US had just plain failed to develop a working nuke, then there would have been a long, drawn out, invasion of Japan. It would have divereted attention and resources away from the re-construction of Europe. Most likely, the Japanese would still have surrendered, eventually, but they might concievably have managed more favorable terms of surrender, perhaps even retained thier existing government after the war.
The fact that the plane dropped a bomb on Japan was a signal the end was near, all it did was save thousands of American GI’s lives and get the war over much sooner. The end result would have been the same, but the woman probably would not of got the vote in Japan without the USA insisting on equal rights for women at the unconditional surrender.