A few hours after the bombing of Hiroshima, the US sent out propaganda/press releases that explained what they had done, how they had done it, etc.
The Japanese high command sent a team of scientists to investigate Hiroshima at this point, to confirm whether the US accounts were accurate and to get a sense of the actual damage.
The Japanese scientific team reported back on August 8th that it was indeed an atomic bomb. On August 8th is also when the Japanese released their first public accounts of the bombing, including casualty reports, through state-controlled media. The US also got its own reconnaissance of Japan at this point, because previous fly-overs had been hampered by the excessive amount of smoke. This was shown to Truman personally on the morning of August 8th, and the photos were soon publicized in American newspapers. The reports of the Japanese accounts appeared in American newspapers by August 8th.
So by August 8th/9th (depends on your time zone) everyone had a pretty good "rough" sense of what had happened in Hiroshima — about 65% of the city area was destroyed, with deaths ranging from tens of thousands to hundreds of thousands. (I have been inexact with the dates here because the time zone issue makes this a bit tricky; Japan is almost 12 hours different from Washington, DC, and is on the other side of the International Date Line. To show how tricky these things can be, and how tight the scheduling was, by the time Truman learned about the death toll at Hiroshima on early August 8th DC time, the strike for the second atomic bomb was starting to get under way, even though it wouldn't drop on Japan until early August 9th local time. None of these time zone issues are nearly as hard as trying to make sense of timestamps from the Potsdam Conference, which was on Moscow war time during the war, to make things easier for the Soviets, I guess...)
One can only do so much surveying and "understanding" in a day or two. After the war it was a high priority of both the US and Japanese was to do comprehensive surveys of the physical and medical effects of the atomic bombs. These took years to complete (and there are still ongoing efforts to assess the long-term biological damage), though the first US survey teams arrived in Hiroshima and Nagasaki within days of the Occupation beginning. M. Susan Lindee's Suffering Made Real is a pretty comprehensive book about US-Japanese efforts to make full sense of the effects of the atomic bombs on Hiroshima and Nagasaki.
The Japanese had very little time to realize the damage done by the atomic bombing of Hiroshima before a second bomb was dropped on Nagasaki, and in fact the bombings coincided with the Soviet invasion of Japanese-held territory in Manchuria. It's not clear at all whether the bombs, or the Soviet invasion, or the combination of the two, are actually what pushed Japan into surrender.
For a longer discussion of this, see these older posts: