It’s an extremely common and persistent myth that Nagasaki essentially exhausted the US’s supply of the bombs, but it’s not true.
Components for the next weapon were being readied to be flown to Tinian and George C. Marschall was told by Groves on the 10th, the day after Nagasaki, that he expected the next bombing to be on the 17th or 18th.
Groves’s basis for this estimate was a telegram from Oppenheimer the day before (i.e. the 9th) stating when the components would leave Los Alamos for Kirtland and thence to Tinian. The high explosive lenses would leave on the morning of the 11th and the plutonium core would leave on the evening of the 12th. There was already a stockpile of the other components, like the casing, on Tinian.
The timing of the two previous attacks had been left to the military, but events were beginning to move rather fast by this stage. Truman removed that existing freedom on the 10th, allowing the military to continue with preparations for another bombing but requiring that the use of this next weapon depend on his explicit approval. Probably independently, Groves and Oppenheimer agreed to proceed cautiously when it came to shipping stuff. Robert Bacher, who had become Los Alamos’s specialist in handling completed plutonium cores, apparently had this core in the car ready to be driven to Kirtland when Groves cancelled the transfer.
Groves’s memo to Marshall is reproduced in John Coster-Mullen’s Atom Bombs (p294 in the 2006 edition I have). It’s discussed in Racing for the Bomb (Steerforth, 2002, p424) by Robert Norris and the exchanges with Oppenheimer are summarised in Critical Assembly (Cambridge, 1993) by Hoddeson et al.
As for bombs thereafter, Stimson was told in a cable from George Harrison on 24th July to expect three more in September, with rate rising to seven in December (Norris, p415). There were overwhelmingly plutonium ones. In the event, this was never achieved, but that was because the Japanese surrender removed all urgency.
However, the notion that there had been only two available does go back a long way - and to a somewhat surprising source. It’s already unambiguously present in Stimson’s famous February 1947 Harper’s magazine article justifying the bombing. This was, understandably, taken as authoritative about such details and the claim repeated on that basis.
(Since the article was heavily drafted by Harvey and McGeorge Bundy, one can always question its reliability on the details. It’s possible that Stimson had completely forgotten asking Harrison about the production schedule at the time.)