Question says the crux.. So many countries are still trying to merely replicate what the US did almost a century ago, this is technological breakthrough on a scale too impressive to imagine! What factors allowed this success to happen?
If someone can suggest one or toe books a layman can pick up to understand this event, that'd be awesome too!
They spent an unreasonably large amount of resources on it, and they just barely pulled it off in time for the war. When you commit the labor of 600,000 people, plus $2 billion in raw cash, to something where you have nobody trying to stop you or inhibit you, and you happen to have a huge scientific, technological, and industrial base to draw on, you can just barely pull it off given the constraints at the time.
Plenty of other countries have pulled it off. That the USSR did so is no real surprise — they lacked some of the advantages of the USA but they were also able to commit the resources. The UK took their time about it and were able to do it somewhat on the cheap. France, China — also not terribly surprising though the Chinese proceeded more rapidly than one might expect given the state of Maoist China. These are all states with major industrial bases, lots of people, lots of scientists.
More tricky are places like India, Israel, Pakistan, and South Africa, which were not only trying to pull it off with much more limited resources on the whole, but were trying to do it in much more secrecy. One measure of this: During WWII, the US had a population of around 130 million. This means that about 0.4% of the total population worked on the Manhattan Project. Or around 1% of the total civilian labor force. Israel had a population of about 2 million in the 1960s. So if 0.4% of their population was available to work on their bomb, that would only have been 8,000 people! Additionally, these countries had to do this work in a world where non-proliferation pressures existed, unlike the USA.
I don't want to diminish the difficulty of what the USA pulled off — building a bomb basically 3 years (it didn't really get rolling until 1942) is a feat no other country has managed, but it was just a combination of doing every possible approach at once (a lot of redundancy) and some luck that enough of those approaches actually worked. It was tremendously risky and they just barely pulled it off.
If someone can suggest one or toe books a layman can pick up to understand this event, that'd be awesome too!
The classic book on the subject is by Richard Rhodes The Making of the Atomic Bomb which won a whole series of awards.