I'm going to go against orthodoxy and say that I think the development of nuclear weapons is far from inevitable, and has a basic probability of less than 50%. Here's what you have to happen in order to get nuclear weapons:
1) Have radioactivity and other phenomena explained by a version of physics that leads to the realisation of the possiblity of nuclear fission. From first principles, with no direct evidence. Remember we had a theory that explained combustion satisfactorily for years until someone built a decent set of scales and proved it was illogical. Someone could easily build an atomic and quantum theory that fits the evidence but fails to predict nuclear fission. Even in OTL the scientific jury was still out up until the Trinity test to some extent.
2) Get enough money and resources to build a bomb. This is hard. Look at Iran today. A medium-sized, middleweight country with a decently sized population and a reasonable university system, and it's still taking them years and years. And that's after other people have done all the groundwork. I think OTL was a strange case of a perfect storm: America, a country both rich yet also in a depression, having a government that wanted big public works projects to create jobs (which is particularly ASB-sounding if you know American history) and a country that wasn't at war yet but knew it might be very soon. (The letter from Szilard signed by Einstein that led to the Manhattan Project was signed by Roosevelt a few days before Pearl Harbour - if he had left it any longer, it's likely it wouldn't have reached the president's desk for months or years with the chaos of WW2!)
3) The engineering expertise. I once had to write an essay on how the Manhattan Project was more an achievement of engineering than physics. The project consumed a ridiculous amount of the USA's resources, in wartime, and the only thing even slightly comparable was the Apollo programme. You need companies with the right areas of experience and you need a setup where those companies are independent enought to have branched out into such areas yet not so independent that the government can't order them to work on this project. You need a military that will take this project seriously and has its own corps of engineers to implement the scientists' dreams.
4) Avoid dead ends. The Nazi atom bomb project focused on trying to make a reactor and then make it go critical, having missed the idea of an impact bomb. This, in a country with all the scientific and engineering expertise of Germany, should illustrate why A-bombs are less likely than people think they are. The USA mostly avoided this because they had so much money, whenever there was an either/or choice of designs in the Manhattan Project and one of them might end in a dead end, they just did both. Again, I can't stress enough that the circumstances for this to happen seem rather unlikely to me. And a project like the Germans', in which they might stake everything on a wrong design...it could easily just be cancelled and then no-one else would touch the atom bomb idea, regarding it as being as half-baked as cold fusion in OTL.
5) Actually drop the thing. The US nearly didn't in OTL because there was a small but real concern that the fission might proliferate and ignite the atmosphere, destroying all life on earth. Besides, the war was nearly over. But there were reports of the Soviets working on their own bomb (which, remember, they benefited significantly from the fact that they had their agents infiltrated into the Manhattan Project) and the Bomb was dropped largely as a national propaganda exercise. If it had been peacetime, well, would any politician sign up to a project that might deliver a wonder weapon but would have a 5% chance of destroying the world?
So...yeah. Basically I think the circumstances that led to the development of the Bomb in OTL were rather special. And let's not forget that there are lots of things which most people think are plausible and doable and have been planned for years and years but have never materialised because of lack of government funding: moonbase, anyone?