The entire point of using the atomic bomb was for its psychological/propaganda value. They could have destroyed those targets without atomic bombs; in fact, they had to make some effort to make sure Hiroshima wasn't destroyed by firebombing before they could drop an atomic bomb on it. So keeping the atomic nature of the bomb secret was exactly the opposite of what they wanted to do. They wanted the Japanese (and the Soviets, to a degree) to know that the United States had a new superweapon.

As for proving it was an atomic bomb, there are two very obvious ways to do this. One is to look at the radiological residues themselves. A freshly-detonating atomic bomb has a different radiological "signature" than reactor-made fission products, as well as any other kind of radioactive material you can make. The other is to look at the blast and thermal damage, which was consistent with a very large point-source detonation in the air. For a few miles around, you could look at damage and flash burns and calculate the angles at which they arrived. There would be no way to fake that on a scale the size of a city. The US, UK, Japan, and the USSR all performed these kinds of measurements after the war was over. During the war itself, after the Truman announcement of the bomb (see below), the Japanese sent a team of scientists to Hiroshima to confirm it was an atomic bomb, and they looked at these kinds of things as well and were able to quickly confirm that it was real.

As for newspapers (and everyone else), 16 hours after the Hiroshima bombing, the White House released a statement with Truman's name on it that announced that an atomic bomb had been used. Shortly after another, longer statement about the creation of the bomb was released by the War Department, and a statement was also released by the British on their contributions. Newspaper had been created prior to the use of the bomb by the Manhattan Project (the organization that made the bombs) with the help of William Laurence, a reporter for the New York Times. These were distributed to newspapers in the US with the intention that they could use them as the basis of their own reporting, or just copy them. The goal was to provide a huge amount of "safe" information that would both make the importance of the bomb self-evident and also prevent journalists from doing too much original research into secret topics. Along with this, a few after the Nagasaki attack, the Manhattan Project also released the Smyth Report, a detailed technical-administrative history of the effort to make the atomic bomb, which had been written during the project itself. So this was a very careful effort.

For more on the latter, see esp. the chapter on the "Publicity" program in my new book (chapter 3), Restricted Data: The History of Nuclear Secrecy in the United States (University of Chicago Press, 2021).