They also had the benefit of having stolen information on what did and didn't work, via spies like the Rosenbergs. Why try what doesn't work if someone else has proved the path before

The USSR (first bomb test: 1949) obtained secret American information from spies and the UK (first bomb test: 1952) had secret information that was intentionally shared by the US. France (first bomb test: 1960) is an interesting case because it developed its program relatively frugally without intentional US information sharing or (as far as I know) spies within the US nuclear weapons program. By the late 1950s there was just a lot more information publicly known that couldn't be hidden again. And today there is nuclear information readily available on Wikipedia that would have been considered top secret as recently as the 1970s, like:

https://en.wikipedia.org/wiki/History_of_the_Teller%E2%80%93...