Why did the United States drop a uranium bomb instead of a plutonium bomb on Hiroshima?

Details below on why a U-235 bomb was used at all. As to why that was the first bomb; because it was ready first and departed the continental U.S. only hours after the Trinity test for Fat Man.

The target and bomb pre-assemblies (partly assembled bombs without the fissile components) left Hunters Point Naval Shipyard, California, on 16 July aboard the heavy cruiser USS Indianapolis, arriving on 26 July. The target inserts followed by air on 30 July.

The Fat Man bomb assemblies (yes, plural) began arriving on Tinian only 12 days later, July 28:

The first plutonium core was transported with its polonium-beryllium modulated neutron initiator in the custody of Project Alberta courier Raemer Schreiber in a magnesium field carrying case designed for the purpose by Philip Morrison. Magnesium was chosen because it does not act as a tamper. It left Kirtland Army Air Field on a C-54 transport aircraft of the 509th Composite Group's 320th Troop Carrier Squadron on 26 July and arrived at North Field on Tinian on 28 July. Three Fat Man high-explosive pre-assemblies (designated F31, F32, and F33) were picked up at Kirtland on 28 July by three B-29s: Luke the Spook and Laggin' Dragon from the 509th Composite Group's 393d Bombardment Squadron, and another from the 216th Army Air Forces Base Unit. The cores were transported to North Field, arriving on 2 August, when F31 was partly disassembled in order to check all its components. F33 was expended near Tinian during a final rehearsal on 8 August. F32 presumably would have been used for a third attack or its rehearsal.

In truth, the uranium bomb was both so straightforward in design and elegant in function that, once the necessary calculations had been made and checked, not even one person at Los Alamos thought it necessary to test it.

Additionally, as of early summer 1945, it was still taking about 6 months to enrich enough U-235 for a bomb. That calculus wouldn't change until an improved enrichment technique was proven in mid-September.

The design of the Fat Man plutonium bomb, on the other hand, was so complex and fraught that many were uncertain, even on the morning of the Trinity test, whether it would work. There was a pool amongst the scientists on the test result, with bets ranging from it being an embarrassing fizzle to sinking into the Earth's core. These bets were certainly, in part, tongue-in-cheek: but the uncertainty about the design was real.

Note from my link above that, at least in part, the point of dropping the Nagasaki bomb was to announce, loud and clear to every knowledgeable person in the World (of which there were a great many): "We solved the uranium enrichment problem. We can now drop these indefinitely about every 7 to 10 days." The Japanese heard that loud and clear.

To respond to some of the comments below: it is an error to think that, because both bombs use fissile fuel, that they were "basically similar". That is the exact opposite of the truth.

Due to the very different spontaneous fission rate of the uranium and plutonium fuel, the Los Alamos physicists realized early on that a plutonium bomb required a vastly different, and more complex, mechanism than the uranium bomb.

Little Boy, the uranium bomb at Hiroshima, was essentially a fancy gun barrel that slid down a track onto a fixed bullet. Other than fixing the bullet and firing the barrel, its a firearm design is fundamentally the same as every firearm of the past half millennium. Provided that an appropriate barrel speed is reached and maintained as the barrel approaches the bullet, the bomb will go off. Minor errors in calculation, or speed, simply modify yield as the fuel—uranium-235—is forgiving in its spontaneous fission rate.

Fat Man, the Nagasaki bomb, uses plutonium fuel. The physics is (early) graduate level, so I won't go into details here: but the spontaneous fission rate of plutonium (specifically, the Pu-240 contaminant) is significantly greater than for enriched uranium. This means that the time period for assembling the fissile fuel must be much much less than for Little Boy. This required an implosion device, where several (16?) spherical sectors are fired inwards simultaneously, to assemble perfectly into a sphere in something like a microsecond. (Possibly even less; I haven't read up on the fine details in some years.) Minor errors in timing or assembling results in the fuel melting before it can go critical, and not reaching critical mass at all. One is left not with the debris of an exploded bomb, but simply a blob of melted plutonium and other bomb materials.

Finally, it's important that recall that a uranium bomb could have been ready in 1944, except that the uranium enrichment process of the time was so slow and inefficient. We should be glad of that - it being the hurdle that both Japan and Germany were unable to overcome in their bomb programs. The importance of even relatively tiny amounts of enriched uranium is why, at the end of the war, Germany tried (albeit unsuccessfully) to get its small supply to Japan by submarine, in hope that it might speed development of a Japanese bomb.

Page 66,67-68 Chapter Two - The Fission Bomb had to Come First (PDF):

Caltech physicist Richard Tolman suggested implosion as early as 1942, but the implosion method ... constituted an extremely complicated shockwave phenomena.
...
Generally an implosion device works in the following way: A subcritical core ... is surrounded by a shell of high explosives -- part of a lens structure that focuses the blast into a converging inward moving front. Electrical charges detonate the explosives nearly simultaneously, so the resulting blast is relatively symmetric, causing an even implosion of the core and compression of the fuel. Due to this compression, the core becomes super critical, and begins to expand outward, causing an explosion.
Modelling these process provided not merely a challenge, but in the summer of 1944 no one knew if an implosion would work at all.

Page 67 Chapter Two - The Fission Bomb had to Come First (PDF):

In the spring and summer of 1944, Emilio Segre's experimental physics group realized that spontaneous fission in Pu240 made the plutonium gun idea unworkable; it would not be fast enough to tolerate the added neutrons. ... A uranium bomb could be made by the summer of 1945, but probably only one. Thus the Laboratory turned to implosion as the only practical means of utilizing the plutonium available in the summer of 1944.

Example of how the scientists and engineers, not the bureaucrats, ran Los Alamos Page 70:

... the army recruited several high school graduates [SED] from all over the U.S. and sent them to Los Alamos. ... [k]nowing nothing about the purpose of the project .... One cycle took about three months to complete until Feynman obtained permission from Oppenheimer to inform SED's about the purpose of the project. Excited about fighting a war, the SED's quickly invented their own programs to speed the effort, and completed nine problems in three months.

Some additional bomb details, with thanks to Vorbis for digging them up:

1. The purpose of the conventional explosives was to collapse it from the less dense delta phase to the more compact - but still solid - alpha phase (allotropes of plutonium are quite weird! nuclearweaponarchive.org/Nwfaq/Nfaq4-1.html)
2. The conventional explosive shell was divided in 32 sections, as a truncated icosahedron (see here: en.wikipedia.org/wiki/Truncated_icosahedron#Applications )

From the first: my imagery above of the Fat Man bomb construction details, and of the details of a fission fizzle, may be off slightly.