No. The concept of a fission explosion had been theorized for some time. The issue was - it would cost a fortune to find out if it would actually work, given the technology of that age. In the 1920's and 1930's, radiation science was still emerging, and most people working with radiation were pursuing peaceful uses, so the idea of an uncontrolled chain reaction was more interesting theory that serious goal. A parallel theory held that plutonium could also initiate a nuclear explosion, but plutonium doesn't exist naturally... it has to be made in a reactor.
Both approaches would require 25-50 pounds of either pure U-235 or Pn. At that time, both were only known to science in quantities of a gram or less, obtained only with great difficulty. It would cost a fortune to obtain that much of either substance. Plus, they had to figure out how to keep the device safe until initiation... not simple.
And, there was the possibility that a fission bomb might just 'fizzle', as in initiating a slower chain reaction that released a lot of radiation, but not the instant and massive release of energy that the theory suggested might happen.
At that time, no nation or organization was going to devote that level of resources, greater than most nations GDP, to investigate a theory that might or might not work as theorized.
The pressing need was WW2, and the fear that Germany and Japan were also developing such a powerful weapon, plus Albert Einstein's appeal to Franklin Roosevelt in 1941. Only a person of Einstein's stature could have persuaded the US of the danger, and the need to fund the project. Only the US at that time had the resources, and was free of attacks on its economic infrastructure that it could spare that level of resources. It was estimated that when the Oak Ridge uranium enrichment facility was operating during the war, it was consuming as much as 1/7 of the total electric power in the US. Only the US had enough suitable metal to make electromagnets of that scale... since copper was needed for wartime, they used the silver in the US reserves to build the Oak Ridge calutrons that consumed so much power.
Once the theory was proven, then making a bomb became just a matter of getting the material and assembling the device. It was taking the risk to fund an unbelievably expensive project that might not work as theorized, that posed the greatest obstacle.
So while the inner workings of a nuclear bomb could be kept secret, the fact that it worked, and the devastating power it unleashed, could not.
The details communicated by Fuchs and the Rosenbergs to the Soviets cut some time off of their bomb project, but the Soviets would have achieved a nuclear bomb even without their help. Unlike the Allies in WW2, the Soviets knew it would work.
Note also that China developed a nuclear bomb in the 1960's with no assistance or purloining of secret information.