Within 24 hours.

It is difficult to find the exact time of the various releases. This press release from the White House states it came out sixteen hours after the attack, or about 12:15 AM on 7 August Tokyo Time. It appears it was preceded by this radio broadcast, though they may have been released together. Later that day Secretary of War Stimson released this press release. Given modern communications, within an hour of Truman's broadcast it would have been known to most of the globe that was awake at the time (including world leaders), and the rest by morning.

How long it took for news to spread in Japan is harder to nail down and varied greatly. The Japanese fought to keep Allied propaganda from reaching their population. Thus, while America dropped some five to six million leaflets and some Japanese-language newspapers with images by 9 August (Translations 1, 2 and began broadcasting a message every 15 minutes, how many Japanese heard the news cannot be known.

But even without Allied news some did know, especially in the leadership. For this I will quote from this 1946 US report:

At 8:16 A.M., the Tokyo control operator of the Japanese Broadcasting Corporation noticed that the Hiroshima station had gone off the air. He tried to use another telephone line to reestablish his program, but it too had failed. About twenty minutes later the Tokyo railroad telegraph center realized that the main line telegraph had stopped working just north of Hiroshima. From some small railway stops within ten miles of the city there came unofficial and confused reports of a terrible explosion in Hiroshima. All these reports were transmitted to the Headquarters of the Japanese General Staff.

Military headquarters repeatedly tried to call the Army Control Station in Hiroshima. The complete silence from that city puzzled the men at Headquarters; they knew that no large enemy raid could have occurred, and they knew that no sizeable store of explosives was in Hiroshima at that time. A young officer of the Japanese General Staff was instructed to fly immediately to Hiroshima, to land, survey the damage, and return to Tokyo with reliable information for the staff. It was generally felt at Headquarters that nothing serious had taken place, that it was all a terrible rumor starting from a few sparks of truth.

The staff officer went to the airport and took off for the southwest. After flying for about three hours, while still nearly 100 miles from Hiroshima, he and his pilot saw a great cloud of smoke from the bomb. In the bright afternoon, the remains of Hiroshima were burning.

Their plane soon reached the city, around which they circled in disbelief. A great scar on the land, still burning, and covered by a heavy cloud of smoke, was all that was left of a great city. They landed south of the city, and the staff officer immediately began to organize relief measures, after reporting to Tokyo.

Tokyo's first knowledge of what had really caused the disaster came from the White House public announcement in Washington sixteen hours after Hiroshima had been hit by the atomic bomb.

Taken together, it appears certain that the Japanese leadership knew Hiroshima had been completely destroyed at some point on 6 August, though they did not know the cause. At some point after midnight (local time), the news it was an atomic bomb spread throughout the leadership, so by the morning of 7 August that too would have been known.

It is reasonably well known that the Japanese had dispatched a team to make a formal report of the devastation, which had not been complete by the time Nagasaki had been bombed. While I have not seen this report or a summary thereof, usually these are detailed and include information on what occurred, what was destroyed (occasionally with basic maps), and what may be salvageable: we can presume this report would be similar in scope.

For example, you can read Secretary of the Navy Knox's on Pearl Harbor here (third PDF, page 29), given to Roosevelt at 10PM on 14 December, with similar initial messages on other attacks earlier in the third PDF. Knox's report cites interviews, radar plots, captured Japanese documents, etc.: the Japanese report on Hiroshima would similarly have taken advantage of all available sources and include similar details.