It's my understanding that the Trinity test (plutonium), Little Boy (uranium-235), and Fat Man (plutonium) bombs pretty much used up the supply of fissile material that we had been able to produce to date.
My goal here is to try and further understand the magnitude of the decision to bomb one city right after another while possibly blowing our entire wad of boom material at the same time.
Ray Monk's Inside the Center: The Life of J. Robert Oppenheimer "Immediately after the Nagasaki bombing the Allies did not possess any more atomic bombs. It is true, as Groves puts it, 'our entire organization both at Los Alamos and at Tinian was maintained in a state of complete readiness to prepare additional bombs', but, as he himself reported to General Marshall, the earliest data at which the next bomb could be assembled for use was 17 August, and almost everybody expected the war to be over by then."
We had a third available for use on the 19th, another three through September (about every 10 days), and then a probable monthly production of three per month starting in October.
Source is a transcript of a telephone conversation between General Hull and Colonel Seaman 13 August 1945. A more easily readable text version is also available.
The third plutonium core would have been available around August 17-19th. It would have been shipped to Tinian on August 12th or 13th, and then it would have taken several days to assemble it into the weapon. Lots of details in the links, including links to various documents on the subject.
The more interesting issue relating to your concern is what would have happened if they had delayed the dropping of Little Boy and instead used its 64 kg of U-235 to produce 8-10 uranium-fueled Fat Man style bombs. It would have delayed things by a week or two but would have increased the available stockpile dramatically. It was proposed by Oppenheimer in July 1945 but nixed by Groves because of the time delay.