Yes, there was a third bomb that would have been ready to go within two weeks of Nagasaki (Aug 19). Let me tell you the story of ...
THE DEMON CORE (/shriek of terror).
Edit - reading error (On August 13, the third bomb was scheduled. It was anticipated that it would be ready by August 16 to be dropped on August 19)
The nukes may have made them feel more comfortable about the PR aspects of surrendering, and so sped things up a tiny bit, but one nuke would have worked as well as two for that, so I think the most likely effect of one of the bombs failing would have been that the surrender would have happened at the same time (or within a day or two) and only one city would have been nuked.
NB the idea that the Japanese could some-how 'reverse engineer' a (failed to properly explode) plutonium bomb during ww2 is ASB = they have no source of uranium and no enrichment plants and no reactor (for produccing plutonium). Nor do they have the knowledge (or chemical plant) needed to seperate plutonium from all the 'impurities'. TallBoy might be easier to reverse engineer, but there was only one TallBay because it took years to enrich enough uranium to the required state ...
Hmmmm....
- It's difficult to dispute the consensus that if one of the bombs was going to fail, it was Fat Man, the plutonium implosion bomb used at Nagasaki, a more complex and less reliable design than the uranium gun warhead Little Boy represented. So it's reasonable to start with this sequence: Hiroshima is bombed on Aug. 6; the USSR declares war on Aug. 8; Nagasaki is targeted on Aug.9 but the bomb does not detonate.
- Against @Protagoras's view, the late James Hornfischer (The Fleet at Flood Tide: America at Total War in the Pacific, 1944–1945) contended that BOTH bombs were necessary to break the war cabinet deadlock, a view shared a bit more equivocally by Richard B. Frank (Downfall: The End of the Imperial Japanese Empire). It is impossible to say with certainty which view is right, but Hornfischer's is the one with potential for a different war's endgame.
- As @Turrosh Mak rightly states, a third bomb (a plutonium bomb) was ready and could have been dropped as soon as August 19. Its nominal target was (tentatively) to have been Niigata, with Kokura as alternate (unless Groves decides to try targeting Nagasaki again). Beyond that, we have a broad understanding of bomb availability, if not quite down to exact dates, thanks to Gen. Leslie Groves' July 30 memo: "In September, we should have three or four bombs. One of these will be made from 235 material and will have a smaller effectiveness, about two-thirds that of the test type, but by November, we should be able to bring this up to full power. There should be either four or three bombs in October, one of the lesser size. In November, there should be at least five bombs and the rate will rise to seven in December and increase decidedly in early 1946. By some time in November, we should have the effectiveness of the 235 implosion type bomb equal to that of the tested plutonium implosion type."
But it strikes me that the OP asks us to consider how each side's decision-making changes if the Nagasaki bomb is a dud:
A) Do the Americans alter their plans after a dud, even setting aside a Japanese refusal to accept surrender terms?
B) How do the Japanese react to a Nagasaki dud?
A) Now, a failed bomb *would* cause some anxious soul-searching in Groves' command, as well as in Washington. Suddenly their ace card looks shakier, not because of fears the Japanese will reverse engineer it (they have neither the expertise nor the time, as
@HexWargamer notes, and anyway the AN 219 destruct fuses would have rendered it an ugly mess on impact), but because Japanese realization that the American a-bombs may be very unreliable, which may risk ironically bolstering the hardliners. The bigger, more immediate problem for Groves and Truman is that the next bomb was a plutonium bomb, as were most of the ones in the pipeline. Do they hold off until they can study what happened with Fat Man? Or do they keep trying? After all, they *had* successfully detonated a plutonium bomb already at Trinity. Thus...when Groves meets with Parsons, Purnell, Spaatz, and Lemay for the postmortem, I think they'll likely assume the failure wasn't the bomb itself, but more likely the
Archie radar antenna - a fixable problem - and so advise Truman.
My sense of Truman's mind at this point is that he will order more bombs be dropped (along with continuation of Lemay's incendiary bombing campaign) until the Japanese communicate some willingness to talk, and hope the next ones work. If the war drags on into autumn, the one constraint which will manifest itself was Marshall's desire to have some bombs (up to 9, if he could get them) for tactical use at the outstset of the Operation MAJESTIC invasion of Kyushu. It is unclear when that would have become a real debate. I think the eagerness to show Tokyo that Nagasaki was a fluke (assuming Tokyo has figured out what happened at Nagasaki) will keep Marshall at bay until at least October.
B) The greater wild card is the Japanese war cabinet dynamic. We are after all talking about not just a scenario where only one bomb is dropped, but rather where a
second bomb is dropped but is a
dud, and that matters because that's greater ammunition for Anami's shotlocker against the peace faction. How quickly will the Japanese figure that out? It will take time to identify and recover the bomb remains and figure out what has happened. How long will that take? That depends on the manner in which it fails. If they do so before the August 19 bomb is detonated on Niigata, that might work against the Niigata bombing's effect on the cabinet and Hirohito. A best case scenario is that we just end up with Hirohito's intervention of August 9-10 pushed back to Aug. 20. A slightly worse case might be that the realization of the Nagasaki dud causes a political holding pattern until more bombs are dropped in September, on top of the cumulative losses racking up in Manchuria.
But the worst case scenario would be that it gives enough impetus to a hardliner coup, be it soft or hard, to keep the war going. At which point, we could well be looking at something like The Red's
Decisive Darkness timeline (though I am somewhat more skeptical about the possibilities of rapid Soviet advance into the Kurils and Hokkaido).
