It just seems that they spent the equivalent of billions of dollars and tens of thousands of man-years, on a highly risky project that they couldn't guarentee would work or be ready during the course of the war. Whilst they had existing stocks of chemical and biological weapons which were cheaper to develop and to manufacture and have the advantage that the buildings are still standing and so it's easier to occupy the area and it the chemical or bio weapon is short lived there's less residual problems then there are with localised fall out.
It's worth noting that at no point did anyone say, "should we develop nukes or should we just use BW/CW?" That wasn't how anyone saw the options on the table in the 1940s. These were considered entirely separate technologies, entirely different questions to ask.
One might ask why that is, but it's hard to prove a null hypothesis. Is it because they saw BW and CW as "marked" categories because of their global prohibition? Maybe. Is it because they thought an atomic bomb would be much more destructive than either of those? Maybe. (In terms of mortality rates, nuclear weapons are hard to beat for a large area.) Is it really a question about why we, in retrospect, tend to lump all three together as "weapons of mass destruction," despite them having huge differences? Maybe. (Many bio and chem warfare scientists in the Cold War thought the lumping was unfair — they thought their weapons were much more humane and precise than nuclear weapons, and deserved to be treated separate from them.)
I think one could trace many different answers here. Part of it is an imagination of what the "nuclear revolution," as it would later be called, would do. They saw atomic energy as opening up entirely new frontiers — the novelty itself became a push for the work to some degree.
But anyway — my main point is, the question you are asking is interestingly not the one that was on the table at all at the time. Bio and chem warfare was considered uninteresting and off-limits (beyond building up stockpiles) — nuclear weapons were not. That we today might see them as similar is the more latter-day assumption, and it is a curious sign of how their attitudes then are not ours today that they do not at all line up on this issue.
Deterrence and unfamiliarity.
Whilst the US haven't ratified the 1925. Protocol on the Prohibition of the Use in War of Asphyxiating, Poisonous or Other Gases, and of Bacteriological Methods of Warfare, colloquially known as the Geneva protocol, until 1970., they respected its prohibitions.
The main reason is the way chemical and biological warfare is and was seen - as something monsterous, inhumane and inappropriate for classical warfare, that defies all rules of warfare.
Bypassing of the 1899 Hague convention during WW1 aside, chemical and biological warfare agents after 1925 was seen as a deterrent - somewhat akin to the way we see nuclear armaments today. The biological agents' possibility of causing a pandemic is seen as especially volatile. As such, all forces fighting in WW2 were aware their enemies had stockpiles of chemical and to a degree, biological warfare agents - and were aware if they used theirs, enemy would have an excuse to use their own in retaliation, which would cause casualties on both side to skyrocket - something both sides, no matter how desperate or filled with bloodlust they were, wanted to avoid.
during the 1943. Italian campaign, the Americans feared that Germany might be on the brink of using chemical agents in Italy, based on prisoner of war interrogations and intelligence passed on by the Italians. On the night of 2 December 1943, a German air raid on the Italian port of Brisi sank several ships, including an American ship loaded with mustard shells that were being deployed for possible retaliatory use. The raid was highly destructive and it was difficult to determine how many of the casualties were from mustard released into the harbor, but mustard casualties likely numbered in the hundreds.
There's also another thing worth mentioning, which is, in the 1940s we didn't understand radiology, radiation and nuclear fallout as much as we do today - the atomic bombs' power was seen in the equivalent of kilotons of TNT, and all 'side effects' weren't as prominent in people's perception as they are in ours today, after 70 years of Cold War and post-Cold War nuclear fearmongering.
Nuclear power was something new, destructive, and as such didn't have a tarnished reputation like chemical and biological warfare. For example, US forces decided to use tear gas instead of mustard gas during the Iwo Jima campaign in May 1945., and we can attribute that to both the fear of chemical warfare retaliation, and the will to have a moral superiority over one's opponent, and all the benefits this brings to the overall military morale.
Sources:
Rick Atkinson, The Day of Battle: The War in Sicily and Italy, 1943-1944, 2007.
Richard Frank, Downfall: The End of the Imperial Japanese Empire, 1999.