Earliest possible use of nukes?

The largest point of divergence I can think of is if someone takes Ida Noddack's paper regarding Fermi's neutron bombardment experiments in 1934 where she correctly criticised his conclusions and suggested the idea of what would be later recognised as fission as a possible solution seriously, she didn't offer any experimental or theoretical proof to back it up though which combined with having mistakenly claimed to have discovered a new element previously saw it be dismissed. That would move things up four to five years or so, whether the following developments would then also be advanced an equal amount of time is another matter. IIRC there are potentially earlier possibilities but going from memory they start to become somewhat tenuous and would require you to begin from a 'scientist X discovers element/process Y because' point.
 
Probably July the 16th 45 when the Trinity device was detonated it was a Fat Man core so could possibly have been dropped. The Manhattan Project was so big and bomb grade U238 or Pu239 was so hard to make I cant honestly see anyway of hurrying things up. The theoretical work was the easy part but building the whole infrastructure of Manhattan Project isnt going to happen any quicker even if you started 4 years earlier the money wouldnt have been available till 1942.

Little Boy cant come any quicker even though it was a simpler design and wasnt thought to need any testing but its problem was The U235 it was harder to make than Plutonium and there wasnt enough for more than 2 Little Boys until after the war.
 
Could an a-bomb have possibly been used before WWII?

I very much doubt it a lot of the theoretical work on Fission and how much you need to start the reaction was done in the mid 30s to early 1940s. Then there was the fuel problem U238 and Pu239 hadnt been isolated outside a lab before 1940 and no one knew how to seperate the desired bomb Isotopes in the required quantities before about 1942. Even then it took the Manhattan project about 3 years to build up enough fuel to build the bombs, it took a while to build the Hydro Electric power stations to fuel the Centrifuges.

One of the reasons the Germans werent going to get a bomb anytime in the war is they didnt have the spare power needed or the copper and concrete to get the power. That and the fact that the Nazi scientists were working with the wrong physics and they had sent the guys who knew how to do it into exile or the gas chamber.
 
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Probably July the 16th 45 when the Trinity device was detonated it was a Fat Man core so could possibly have been dropped. The Manhattan Project was so big and bomb grade U238 or Pu239 was so hard to make I cant honestly see anyway of hurrying things up.

Probably not possible to speed up the Manhattan project itself, but it's not beyond the realms of possibility that it could have been started up to a year earlier.

Key dates and feasible PODs:
  • Einstein-Szlilard memo on feasibility of nuclear weapons delivered to Roosevelt 11 Oct 1939. FDR sets up Advisory Committee on Uranium which meets for first time 10 days later.
  • Frisch–Peierls memorandum on critical mass delivered sometime during March 1940 and brits set up MAUD committee 10 April 1940, with a side-channel to the US.
  • Franz Simon writes a MAUD report on isotope separation in Dec 1940 leading some to conclude nukes are "inevitable"
  • MAUD memo on critical mass, fast fission and bomb design sent to Uranium committee March 1941, locked in safe and ignored
  • MAUD committee issues detailed report recommending on gun nuke and gaseous diffusion enrichment 15 July 1941, advance copy direct to US
  • Brit military decide to go for nukes 3 Sep 1941
  • US gets 'official' copy of MAUD report 3 Oct 1941, 3 days later FDR asks for a letter to be drafted to ask UK for more info.
  • US enters war 7 Dec 1941
  • S-1 Uranium committee dedicated to US nuclear weapons development has first meeting 18 Dec 1941, get formal approval from FDR 1 month later
  • "Manhattan Engineering District" set up 13 August 1942 and gets its top priority blank check on 23 Sept 1942
So a very plausible POD (Vannevar Bush has a conniption fit when he sees the early copy of the MAUD report, goes straight to FDR and tells him There Is A Problem) kicks the US peactime effort into higher gear six months earlier than OTL. The earlier info from this means the program could perhaps go straight into crash-priority at the start of the war, starting the Manhattan project 9 months earlier than OTL.

Less aggressively, you could have the Brit military decision leaked via Churchill and move the S-1 setup to be at the same time, shaving 3 months there.

More aggressively, you could have the Advisory Committee on Uranium be full of piss and vinegar and physicists, moving ahead as fast or faster than the British effort but starting 4-5 months earlier. Again this probably gives you a military programme running for several months in peacetime, building up expertise, resources and organisation with a weaponisation 'go' decision in the first half of 1941. The infrastructure would then build up along with the rest of the US mobilization effort for almost six months and then accelerate further to 'discretionary AAA priority' from the DoW. Fold the UK efforts in as complement and maybe a deliverable nuke in mid 1944 along with a production plant for a bomb or two per month from there on?

Apologies for the mega-quote from mother of all dubious sources, but it covers a lot of the background. From the entry on Lyman Briggs:
In 1939, President Franklin Roosevelt called on Briggs, by then aged 65, to head "The Uranium Committee", a secret project to investigate the atomic fission of uranium, as a result of the Einstein–Szilárd letter. Even though Roosevelt had sanctioned a project, progress was slow and was not directed exclusively towards military applications. Eugene Wigner said that "We often felt we were swimming in syrup". Boris Pregel said "It is wonder that after so many blunders and mistakes anything was accomplished at all". Leó Szilárd believed that the project was delayed for a least a year by the short-sightedness and sluggishness of the authorities. At the time Briggs was not well and was due to undergo a serious operation. He was unable to take the energetic action that was often needed.

Meanwhile, in the United Kingdom German refugees Otto Frisch and Rudolf Peierls made a breakthrough, indicating that it would be possible to make a bomb from purified U-235. From June 1940, copies of British progress reports were sent to Briggs via a British contact in Washington, Ralph H. Fowler. In March 1941 a British committee of Nobel Prize–winning scientists, called the MAUD Committee, concluded that an atomic bomb was "not only feasible, it was inevitable". They also pointed out that a large part of a laboratory in Berlin had been devoted to nuclear research. A copy of the MAUD Committee's interim report was sent to the Briggs in the USA because Britain lacked the resources to undertake such a large and urgent program on its own. Britain also wished to move its key research facilities to safety across the Atlantic. The MAUD Committee issued another report giving technical details on the design and costs on 15 July 1941.

Britain was at war and felt an atomic bomb should have the highest priority, especially because the Germans might soon have one; but the US was not at war at that time and many Americans did not want to get involved. One of the members of the MAUD Committee, Marcus Oliphant flew to the United States in late August 1941 in an unheated bomber to find out why the United States was ignoring the MAUD Committee's findings. Oliphant said that: "The minutes and reports had been sent to Lyman Briggs, who was the Director of the Uranium Committee, and we were puzzled to receive virtually no comment. I called on Briggs in Washington, only to find out that this inarticulate and unimpressive man had put the reports in his safe and had not shown them to members of his committee. I was amazed and distressed."

Oliphant then met the whole Uranium Committee. Samuel K. Allison was a new committee member, a talented experimentalist and a protégé of Arthur Compton at the University of Chicago. "Oliphant came to a meeting", Allison recalls, "and said 'bomb' in no uncertain terms. He told us we must concentrate every effort on the bomb and said we had no right to work on power plants or anything but the bomb. The bomb would cost 25 million dollars, he said, and Britain did not have the money or the manpower, so it was up to us." Allison was surprised that Briggs had kept the committee in the dark.

Oliphant visited other physicists to galvanise the USA into action. As a result, in December 1941 Vannevar Bush, director of the powerful Office of Scientific Research and Development, undertook to launch a full-scale effort to develop atomic bombs. As the scale of the project became clearer, it came under direct military control as the Manhattan Project.
 
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