What if the Manhattan Project is begun over two years earlier and is able to produce an atomic bomb in 1943 and not 1945?
When in 1943?
Let's say exactly two years earlier - 6 and 9 August. Mussolini is gone, and the Allies are closing in on Messina in Sicily. Hamburg has been destroyed by firestorm. The Soviets have just counterattacked at Kursk and driven across the German start line. In the Pacific, U.S. forces have just taken New Georgia in the central Solomons, Australians are fighting for Salamaua and Lae in New Guinea, and the British Arakan offensive in Burma has failed.
The Allies use the Bomb on Berlin. They probably don't get Hitler with the first one - it will be dropped at night, and it would take a pretty close hit to cave in the Fuhrerbunker.
The second and third Bombs? Koln and Hamburg have been savaged already by Bomber Command. The second Bomb - probably the Ruhr. The third Bomb: Bremen? Frankfurt? Kiel? Munich? Deep targets would be avoided except Berlin.
These are hammerblows. The handwriting is on the wall. Hitler and his Nazi cronies will reject surrender, but almost everyone else will want to give in ASAP. That means general Army support for a Schwarze Kapelle coup d'état. The war is over by the end of August.
(Italy, the only other possible target, will be frantically signalling its intent to surrender. The other Axis powers will also surrender.)
There has been no Teheran Conference to allocate occupation zones. Things will be rather ad hoc, with Hungary, Romania, and Bulgaria hoping to refuse entry to Soviet forces, which are still hundreds of km to the east. It's possible that Churchill will press for US/UK occupation, with a Soviet minority role, and that FDR will go along. The USSR is in a far weaker position, and the US position is rather heady.
Japan will freak. At this time there is no way to Bomb Japan, but it won't take long for the concentrated power of the US/UK to get one. Though OTOH, at this time the USN carrier fleet is very limited. Four of the six pre-war carriers have been sunk; the only new carriers are the first three
Independence class CVLs.
It wasn't until November 1943 that the Allies attacked a Japanese island by purely seaborne force (Tarawa in the Gilbert Islands).
But the Allied war effort in the Pacific is just getting into gear. In the 20 months after Pearl Harbor, the Allies basically only checked the Japanese advance and made very small inroads against them. The next few months saw additional minor progress in New Guinea and the Solomons.
Then in the next 19 months, the Allies swept forward, gutting the Japanese Empire. A year after the first landing in the Gilberts, the US invaded the Philippines, 6,000 km away.
Having the Bomb should be useful, but how?
There aren't many useful targets that aren't also occupied Allied cities - one wouldn't Bomb Rangoon or Singapore, or even Rabaul. Truk is about 2,000 km from the nearest Allied base - about as far as England to Romania, or Tunisia to Ukraine.
One might see Bombs used to clear island targets for invasion; the radiation hazard being set against the often-horrendous casualties from conventional attack. (E.g. Tarawa, Peleliu, Iwo Jima...)
The Allies can now flood the Pacific and SE Asia with troops and land based aircraft, but can they compensate for the lack of carriers?
A wild card here is that Stalin, somewhat thwarted in Europe, may look to Asia for opportunities. The USSR can apply massive force directly against Japan in Manchuria, which might be decisive. Thus where the US might claim credit for zapping Nazi Germany with The Bomb and winning that war, the USSR might end up with primary credit for defeating Japan.
There is another possibility: if the USSR enters the war against Japan, Vladivostok is less than 1,000 km from the Home Islands, easy range for a Bomb-carrying plane. But would the USSR agree to host a US atomic-bomb squadron when the US is not sharing the technology? And would the US trust the USSR not to grab any Bombs?
The war ending sooner saves many lives and much destruction. A million or so Jews spared from the Holocaust.
The war ending sooner greatly reduces the aggregate US war effort. Mobilization wil reach about the same level, but casualties and combat veterans much reduced. The prestige of the Air Force will be colossal; they will have Won the War in Europe, with everyone else's efforts pushed aside by comparison.