Here are a few more facts that can help address the question of whether it would have been wiser to pursue gun-type U-235 bombs versus breeding plutonium for a bomb capability as soon as damn possible. If one can trust Wikipedia at all:
1) From this table of
energy densities, I infer that a "kiloton" of TNT would be about 4.6 x 10^12 Joules, that is 4.6 TeraJoules, since the table is in millions of Joule per kg.
2) From higher up the table, the energy density of "Uranium-235 used in weapons" is 144 TJ/kg, it would then follow that completely fissioning 32 grams of U-235 ought to release a KT yield.
3) However, this page on
the actual strikes on Hiroshima and Nagasaki gives "Little Boy's" actual yield as 16 Kt, and remarks that that is 67 TJ, so actually a "kiloton" is defined as more like 4.2 TJ, some 9 percent less, perhaps because of a different formulation of TNT or because in practice TNT falls short of its theoretical chemical energy release by some 10 percent. By this definition, it would require some 29 grams of completely fissioned U-235 to release the standard KT.
4) the same entry also remarks that the gun design actually fissioned only 1.7 percent of its material, from which we can infer:
5) 464 grams fissoned; and
6) 27.3 kilograms of U-235 were installed in Little Boy.
Note that
this link gives much higher mass of uranium--64 kg at 80 percent enrichment for 51 kg of pure U-235, and a lower percentage of yield, at just 1 percent.
So in fact the more pessimistic number seems reasonable; we need 64 kg of U-235 to get 16 KT yield with this design. Probably relatively easy tweaks can raise the yield per kg.
The vast majority of the 64 kg of 80 percent pure U-235 was scattered without fissioning, and represented nearly the entire output of the entire uranium enhancement program over all the years it was underway.
Going on, the Nagasaki "Fat Man" bomb's yield was 21 Kt, and according to the page on the bombings contained only 6.2 kg of plutonium! It would seem then there is a factor of ten and more difference--more like 13.5. By the above lower figure for grams of uranium needed to produce 1 KT, we can estimate that the same yield would result from fissioning 610 grams--that is, of U-235. However, the plutonium was surrounded by a U-238 tamper/reflector--it is the momentum of this greater by far mass of somewhat fissionable material that, combined with its neutron-reflecting properties, allows just 41 percent of bare-metal critical mass to reach criticality. And the sources remark that "up to" 20 percent of the total yield is produced by secondary reactions in this relatively inert containing material. So actually, if this were the case at Nagasaki, one has a U-235 equivalent fissioning mass of 508 grams, with another 102 gram equivalent set of reactions happening in the tamper. I haven't yet found any figures for the exact mass of the tamper, but simple geometry suggests it would mass an order of magnitude more than the plutonium, some 400-1000 kg.
The sources say that only about 1/5 of the plutonium is estimated to have fissioned, which is to say 1240 grams--from which one might conclude that successfully fissioning U-235 is more energetic than plutonium, by a factor of nearly 2.5! This does point toward developing an implosion type uranium bomb--but one would have to be an expert on nuclear weapons to declare whether or not it is possible to get the same efficiencies at a given state of the art in a uranium design than in a plutonium one. All I know for sure is, the Manhattan Project had a plan for making an implosion plutonium bomb, but not one for U-235
Now it is not clear to me whether unprocessed pure natural chemically refined uranium would do for the tamper, or whether it is actually necessary to deplete it of U-235 somewhat to get the right reactivity and avoid a fizzle yield. In the former case, the implosion bombs stand a good chance to be a lot cheaper than the gun-type, because one is using an order of magnitude less high-grade fissionable material. Note though that the gun design is a lot lighter on auxiliary stuff; Fat Man had to be designed to fit within a highly modified B-29, and was just under 9.1 metric tons, but clearly only a small fraction of that was heavy metal, and of that only a small fraction was fissionable. Most of those 9 tons had to be chemical explosives to implode it. Clearly the gun design allowed for a bomb that was overall lighter and more compact, albeit rather long.
If it is necessary for the outer U-238 tamper to be depleted of U-235, well that was no hardship for the historical OTL Project, for the heroic efforts undergone to get less than 100 kg of fairly pure U-235 would leave very large stockpiles of depleted uranium around, which could be reclaimed by the implosion PU bombs.
But what we are trying to do here is see if there was a path for the first A-bombs to be available sooner; if we are going with focusing on the implosion design for its much superior efficiency, it is important to know whether we must nevertheless run natural mixes of the different uranium isotopes through some sort of isotropic mass separater to get depleted uranium. If so, we can probably use the enhanced stuff we remove in a breeder reactor to generate the PU. But we still have to make separation plants for the uranium, in addition to reactors and chemical separation we need for plutonium.
Finally, although we can perhaps get higher yields for a given input of unrefined uranium, we still need a good fraction of a critical mass of U-235.
It would appear that it might be correct that there could have been more bombs sooner if the project had concentrated on implosion and bypassed the gun design, but while that is clear in hindsight, how likely would any of the planners be to bet the whole farm on that one unproven design, and would it be possible to be as successful with a uranium core as with plutonium?
If in fact natural undepleted uranium is good enough for the tamper, then a gamble on developing only plutonium processing might pay off. The question is, could say 50 extra kg of refined plutonium have been available in addition to the 6.4*3, or about 20, we can infer was much of the total production of Pu by the time of the attack on Nagasaki? That is, we know there was one spare, one tested at Trinity, and Fat Mat dropped on Japan; 50 kg is how much actual pure U-235 was in Little Boy, so if producing one kg of Pu from X kg of uranium mined and chemically refined is equivalent to producing 1 kg of U-235, perhaps the rate of production of Pu could have been 3.5 times faster. If in fact it was 10 times harder to produce a gram of U-235 then we might have not 7/2 but 30 times the rate of Pu production. OTOH maybe the reason there was only some 20 odd kg of Pu for weapons in 1945 was that difficult as producing U-235 was, producing Pu was even harder, per kg, and totally abandoning all effort to obtain U-235 might just lead to 40 kg of PU total.
And would it come at twice the pace, pushing back the date the first test article could be assembled, back before V-E day, so that a second and third could be ready to use on Europe's western front in the winter or early spring of 1945? Or earlier, in autumn or winter '44? That would depend on the date at which the batch that was used in spring '45 OTL was started. I think also that the way plutonium is generated, it is done in batches, unlike the steady slow trickle of U-235 accumulation. One makes a reactor core, and runs it for a time, then shuts it down, takes all the residue of the core, dissolves it chemically, and chemically extracts the plutonium. If there is a way to run it as a continuous flow, that won't be the first design. It could be that instead of getting 6.2 kg ready much earlier, that same 6.2 is available the same day it was OTL, only now it is accompanied by 35.2 more instead of just 15.2. In that case, the effect of doubling down on plutonium implosion would be not to accelerate the Trinity test at all, but rather to leave Truman with 3 spares instead of 1 after Nagasaki--3 not 4, because Hiroshima would also have been a Fat Man drop instead of using up the surplus Little Boy.
IMHO, no big mistakes were made in the Manhattan Project. It took as long as it took, yielded the product it did, and any rival program pioneering the first A-bomb would have had to plan on investing as much and taking as long for the same results. We know how in hindsight it might have gone faster, but this was not known in advance, and so perhaps--if we know many details, some of which might still be hidden under a cloak of secrecy, of the exact processes needed--we might describe a low-probability time line in which the project leaders took foolhardy chances on wild intuitions and got lucky with earlier, cheaper success. And bear in mind, since they had no rational way of knowing this path to success in advance, that a larger number of TLs involve equally bold risks being taken that do not pay off, leaving the Project up a blind alley, without having "wasted" resources on pursuing the lines they should have to come to a successful result. And all of these TLs taken together are a low-probability subset on the fringes of TLs where the Project management was more sensible and spread resources around to cover many contingencies, some of which pay off--in about the same time frame as OTL.
In order to talk about using A-bombs on the Third Reich, we must either be in one of those improbable ATLs where the American directors took a stupid chance on concentrating on one path they could not know would work, and it nevertheless did, and even then the pacing of various items might not allow for much acceleration. Or we are talking about one where the Axis has done better, has lasted longer while the Allies are behind schedule relative to OTL.
Berlin by the way is clearly a sensible target--if one can get to it! But it was only late in the war that bombers achieved the necessary range and fronts advanced far enough to allow forward bases close enough. And any aircraft headed to the German capital along the known shortest routes from Allied bases would be running a gauntlet of determined air defense. Not so the attacks on Hiroshima and Nagasaki! Even if Tokyo had been chosen as first or second target instead, the B-29s approached over open ocean, over waters the Japanese were too devastated and exhausted to patrol effectively; what air defense was attempted was a point defense. Berlin in a Reich still standing high enough that the city has not already fallen to the Soviets is going to have point defenses superior to those the Japanese could manage late in the war, and an airplane headed there will first have to fly over all sorts of other well-defended potential targets. This is bad enough in terms of bomber attrition, such that few crews survived 20 missions. But when one only has a literal handful of bombs, can one take the risk that the single airplane that is carrying it will not be one of the ones shot down before it reaches the target?
I suspect that if A-bombs are available on the Allied western European front, they are going to be used somewhere else first, despite the obvious desirability of taking Hitler out. Someplace like a major U-boat base, or a strong knot of army forces concentrated, or maybe Hamburg or Bremen. Too bad this will tip Hitler off to hide from any B-29s his people detect.