I understand the project was secret but when the bomb was completed I read that the US government released vague pamphlets declaring that the bomb advertising the bomb was sleek with no military markings. In terms of actual facts american public was given very little information on the Manhattan Project .
After the use of the bomb, the US War Department released a huge amount of material about the weapon, how it was made, the basics of how it worked, and descriptions of the history. It was a major effort that was dubbed "Publicity" by those inside the project, and developed in secret along with the bomb. The goals were to both inform the US, Japanese, and global publics about these new weapons, as well as to make sure that the "narrative" about the bomb was properly "controlled."
The elements of the "Publicity" program were, in short:
Press releases sent out under the names of the President and the Secretary of War, which were the first piece of information released.
Dozens of news stories written by an embedded New York Times journalist (William L. Laurence), which could be re-used by any other journalists for their own stories.
A technical history of the project written by the chair of the Princeton Physics Department (the Smyth Report), released 3 days after the Nagasaki attack.
A Manhattan Project Public Relations Organization which asserted itself as the centralized "clearing-house" for all further inquiries into the project and worked to shape how it was talked about, including attempting to refute rumors and encourage secrecy.
The "Publicity" campaign did give out a LOT of information, but it was very selective. So it emphasized the stories of the scientists who built the bomb, and not the effects on the cities that were hit by it, as an example. There was much that was deliberately and often silently omitted in the name of secrecy and preserving the "proper" narrative. So Americans did know some things but not all things. The control by the War Department was not absolute. So some competing "narratives" were also put out — such as theories about the long-term dangers of radiation (which the War Department attempted to refute with its own scientists), and information from the Japanese about the extent of the damage. But the "official" line dominated the coverage.
As time went on, there came to be more competing sources of these stories. Most Americans had a very poor understanding of the effects of the bombs on the Japanese, for example, until August 1946, when John Hersey's Hiroshima came out, first as an issue of The New Yorker and later as a stand-alone book.