Ouch.
OK, so making a uranium bomb is a matter of chemical engineering (at its simplest) - dissolve the uranium in hydrofluoric acid, boil off the uranium hexafluoride, then use a series of gas centrifuges to extract the U-235 hexafluoride, then recover the uranium by passing the hot gas over something more electropotent like sodium. Then you make the Little Boy gun-type bomb, and hey presto! you're an
atomic playboy.
The catch is, that's not cheap to do even now, and it doesn't get less expensive in 1933. The cost for Pakistan to develop a bomb was estimated in 1965 to be $150m - that's about $1.1 billion in 2013 money. And that's on the cheap side, because it's following a proven development path using the technological advances of two decades of other people's trials and errors.
The Manhattan Project cost ~$2bn in 1945, or about $1.57bn inflation-adjusted to 1938 - and that's without any additional costs for ancillary technologies or techniques in metallurgy or chemical process engineering that might have been used for Trinity but which were developed as part of separate programmes. It also discounts the cost of any research brought in from Tube Alloys, the UK bomb project, so let's just accept that the figure is probably a bit on the low side.
If that was evenly distributed across six years' development time (and it should not be - it should be front-loaded to reflect the need to acquire sites and construct facilities)... well, if you want to spend $560m in 1933 on your bomb project, you have to consider that the entire federal defence budget (excluding veterans' costs) was $600m. That's more than 90% of the cost of the entire 1933 military.
Now, to be fair, we can say that we'll only go for the uranium weapon, which doesn't need reactors to make plutonium like Manhattan did, just lots of centrifuges and some nasty chemicals. Even so, if we halve the costs, I think it's still an expense that's only affordable if there is a clear and present danger to the Union that demands that the US
organize and measure the best of its energies and skills, as JFK later spoke of the Apollo programme.
So, what requires that? Basically, a cold war.
Have a POD that makes WW1 turn into a damp squib, but continue the armament process, and either put the US on the Central Powers side (so they are looking at fighting Britain) or have Mexico there (cf Zimmerman telegram) which is probably better.
That probably requires that A-H and Russia sort out their many and varied problems to become rather richer and more stable. Italy and the Ottoman Empire are tricky and may vacillate between the
Entente and CP in the course of the teens and 20s, while Japan may have a small war with Germany that corresponds with its WW1 goals and then start to pull away from Britain as OTL.
So, A-H and Germany are trying like the dickens to modernise the Ottomans and Mexicans, while the French do the same with the Russians, the Italians are champing at the bit to get their hands on an empire (though they're probably not Fascist without the grim casualties of WW1), and the Brits are thinking that now that the Germans are spending less on navies and more on arms to Guadalajara things can settle down a bit as long as they have a really really big fleet.
There won't be a Great Depression, per se, but the 20s won't be a boom either, as so much money goes on military budgets.
OK, so Szilard comes up with the chain reaction in 1933, and things move on more quickly in the Cold War environment - the Admiralty starts ur-Tube Alloys quickly, and the US and Russia gets wind of it within a month, and the French and Austrians have penetrated the Russians, and the Austrians share it with the Germans. Before the year is out, the American, British, Russian, and Central Powers all have uranium bomb projects underway.
It's unlikely that there will be artillery that can fire it - but a stripped-down B-17 probably can. Not recommended for high-threat environments - but in our decades-long cold war ATL, strategic bombing will come to prominence as the consequences of the bomb programmes are considered, and certainly a heavy airframe capable of delivery can be developed by all the major powers at the time.
