How many atomic bombs did the US have on 5 August, at the beginning of its atomic bombing campaign of Japan?

Short answer: One.

Long answer:

This may seem surprising, but the entire atomic bombing operation was put in a terrible rush as soon as results of the Trinity Test reached Truman at Potsdam on 21 July. On 24 July Truman, after meeting with his right-hand man James Byrnes and the Joint Chiefs, gave the first formal authorization of the atomic bombing of Japan, but for an operation of such magnitude the order was extremely vague. There was no stipulation on the number of bombs to be used, the interval between bombing, or even particular targets; the only clear restriction was that the first bombing to not take place until after Truman had left Potsdam. As a result, everything was left to two organizations — the Manhattan Engineer District (MED) and the Army Air Forces. Maj. Gen. Leslie Groves from the Army Corps of Engineers, head of MED and the entire atomic bomb program: "As far as I was concerned, [Truman]'s decision was one of noninterference — basically, the decision not to upset existing plans." It should be noted that there was never a single, official document outlining the "existing plans." There was a list of recommendations by the Target Committee, a study group created by Groves in early spring 1945, including the infamous list of target cities (i.e. Hiroshima, Kokura, Niigata, Nagasaki), but those were only recommendations. Quoting from Sigal himself: "That decision [by Truman] set in motion the use of as many atomic bombs as the MED could produce as fast as it could ship them to the Pacific until someone qualified or countermanded standing orders" (212).

A side-note: there were in fact two simultaneous, separate production lines for two different types of atomic bombs, a gun-type U235 bomb and an implosion-type Pu239 bomb; "Gadget", the bomb successfully tested at Trinity was of the latter. The first gun-type bomb, later known as the "Little Boy," was shipped by USS Indianapolis and arrived at Tinian on 26 July, but issues with assembly and weather delayed the actual bombing until 6 August. After Hiroshima, with the absence of any order to pause the operation, preparation for the second bomb immediately followed. In fact, Groves did everything in his power to deliver the second bomb, the implosion-type "Fat Man", ahead of schedule. Personally I could not find the exact date "Fat Man" was finally assembled on Tinian and ready to be used, but it was definitely after 6 August. This is evident in the string of commands from Groves asking the scientists to advance the schedule — originally set for 20 August, first to the 11, then the 10, and finally 9 August. The second bomb was originally targeted for Kokura, but local weather forced the pilots to go instead for Nagasaki. And everything after was history.

Source: Leon V. Sigal, Fighting to a Finish: The Politics of War Termination in the United States and Japan, 1945 (Cornell Studies in Security Affairs).

It depends on what you mean by "available for use" but the USA had essentially assembled the uranium-powered, gun-type Little Boy bomb by July 31, 1945. That meant they had all of the non-nuclear components ready to go, and the nuclear fuel of the weapon on hand at Tinian (the final piece of the latter had arrived the day before). They could have dropped the bomb anytime after that point — they were delayed first by the fact that they were supposed to wait until the Potsdam Conference was over (August 3rd) and then by weather issues. A good weather forecast led to the bomb being dropped on Hiroshima on the morning of August 6th (local time).

They also had, on hand, many non-nuclear parts for both gun-type and implosion bombs. They had many "spares." But without nuclear fuel, they were not armed weapons.

The plutonium for the second atomic bomb (the implosion bomb) arrived at Tinian on July 28, 1945. But it took time to assemble the much-more complicated weapon. The original plan had been to drop it a week after the first bomb, but the pushing back of the initial drop date, and further weather forecasts, caused the people on Tinian to think that it would be better to push the schedule up to August 9th. So they started final assembly of the Fat Man bomb on August 7th, 1945, and finished it late on August 8th. This was a "round the clock," rushed job that was done much faster than it was intended to do.

After that — the US had non-nuclear components of bombs on Tinian but now no fuel. The next plutonium hemispheres would have been ready to send out around August 12th or so, and would have been ready to use on August 17th or so. They were never sent.

Throughout this period, even after V-J day but before the occupation of Japan began, the Manhattan Project continued to produce nuclear fuel for the bombs in case it was needed. The rate of production at that point was about than 3.5 bombs per month (3 plutonium cores plus 1/2 of a uranium core). So that is where the projection that 7 bombs could have been produced through October comes from. There were also design changes that were considered at the time that could have about doubled that number (if they stopped making gun-type bombs and instead used plutonium-uranium "composite" bombs). None of these weapons were assembled, however; they were parts that could be assembled (with some effort).

You should think of the Manhattan Project not as just an effort to produce a bomb or two, but as the effort to make an atomic-bomb production system. Once they had the first ones they could continue producing bombs at a regular pace.

The knowledge of the US production capability and schedule was indeed one of the bigger secrets at the time, and afterwards.