I remember hearing several times that the bombs dropped on Japan were a lot deadlier than expected. What were the original projections for casualties for both and how much less than the actual numbers were they?
I found Vincent C. Jones' 1985 book Manhattan, the Army and the atomic bomb. While it doesn't mention the death toll estimates, it does offer some insight on pages 531 and 532. In particular, it quotes Oppenheimer with the words:
the neutron effect . . . would be dangerous to life for a radius of at least two-thirds of a mile.
So Oppenheimer estimated the contaminated area with roughly 3.5 km². Also, if you look at the further discussion of the psychological effects and choice of targets ("a vital war plant . . . surrounded by workers' houses") - I think that it makes clear that the atomic bomb was considered to be just like a conventional bomb, merely with more destructive power. It was expected to completely destroy the plant and damage everything close to it. The expected radius of destruction couldn't have been more than half a mile (meaning an area of 2 km²).
Now let's compare to what actually happened. This military study reconstructs the events and cites a radius of total destruction of 1 mile for both Hiroshima and Nagasaki (meaning an area of 8 km²). It also mentions the contaminated area measured by Americans when they arrived there: 8.9 km² in Hiroshima and 1.5 km² in Nagasaki (with a note that the nuclear fallout in Nagasaki was mostly outside the city).
This document from 1946 essentially estimates the immediate casualties in Hiroshima to 70,000-80,000 people. Given the difference between estimated and actual destruction radius, the expected death toll most likely wasn't more than 20,000 people.
The information on Nagasaki is less definitive. I better link to Wikipedia, you can go through the sources yourself. The numbers here are somewhat smaller than with Hiroshima, probably by factor 1.5.
As to radioactive contamination, it is very hard to say anything definitive here. It is clear that Oppenheimer underestimated the contaminated area. It is also clear that there was no knowledge on the long-term consequences of radiation exposure in 1945. The information that we have now comes to a large part from studies performed in Hiroshima and Nagasaki. I'm not sure whether reliable numbers exist but this article lists another 36,000 deaths in Hiroshima until the end of 1945, significantly declining after that.
Actually those that claim they had a bit of an idea seem to not have read the notes of the targeting meeting.
The primary selection of a target had to have certain requirements and the first one being psychological - they wanted to end this war quickly.
The fact is that the United States had three such weapons at their disposal at the time and wanted to use them in quick succession to give the idea that the United States would use them one after the other until Japan capitulated.
Kyoto was selected as the first target because it was a scientific and cultural community where the psychological effects and the power of the weapon could be understood and communicated to those who would bring an end to the war. That all sounds well intentioned to use it this one time and get the attention of the Japanese leaders. The only problem with the logic here is the idea that there would be intellectual survivors that could inform the Japanese government and tell them to surrender. Think about that logic for a moment they thought there would be plenty of survivors of this first use of the weapon. So the casualty figures had to be very low.
Further Kokura was another target selected and the main target was a complex approximately 4100' long and there concern was to make sure they dropped the weapon precisely on that target so as to destroy it. Again the logic here that they had to be 'on target' with the drop in order to destroy the facility. Again this shows they did not understand the destructive power of the weapon at all.
Given this criteria and other information from the targeting meeting notes one could suspect that these people thought hey just a bigger firecracker not a completely destructive weapon.
Gathering as well from various sources a number of 20,000 was a plausible number of casualties. However one of the criteria for the target was city size and population figures since as strange as this might sound they figured on studying the effects while they were at it. Certain detonation points were selected for this purpose as well, knowing how many miles out from ground zero they could take readings and measurements. I guess if you are going to do it anyway you may as well know what you have and they used some of this data from the Hiroshima blast to determine effects at Nagasaki; however Nagasaki was limited due to the graphical topology and the fact the bomb was dropped almost 2 miles NW of the intended target location - which spared half of the city.
Kokura was the back up target for Hiroshima and they had managed to get just enough of a visual view to drop the bomb on Hiroshima (one of the requirements that was placed on the missions) no use of the radar bomb sighting.
Kokura lucked out on the first mission that Hiroshima cloud cover diminished for it to be bombed.
The second bomb Fat-Man Kokuro was the primary target it had a large military complex and chemical weapons factories - which were known about by the United States by the early summer of 1945.
When Bock's Car arrived in Kokura it was almost one hour later than it should have been and by this time the report of 3/10 cloud cover had changed to 7/10 and the requirement for visual target id caused Kokuro to be aborted and that the secondary target of Nagasaki be attempted - Nagasaki they assumed would have fewer casualties just because of the topology.
So now there is a term people use.. they might say you have Kokura's Luck when you dodge a big disaster.
Nagasaki was bombed but the target point was about 2 miles off. The crew had poor visibility, and was low on fuel and also was pressed into the mission even though the plane was not in ideal conditions. Originally the 'Great Artiste' was to drop the bombs on August 11th but the mission was moved up to the 9th and therefore the Bock's Car was used for the mission. Taken these things into consideration and looking at the crew logs for the flight it seems to me that they really did not have a clear visual of the target - if they had they would have been at least with in 1/2 to 1 mile of the target or they had to have some extreme winds blowing the weapon off course (unlikely).
The following is just my opinion relating to Nagasaki..
Given the pressures to get off the ground on the 9th instead of the 11th, to make the mission even with a bad fuel pump that it was determined the bomb will be dropped somewhere regardless; details about handling of the weapons in the event of a scrubbed mission left out for brevity. Given Bock's car loitered at the rendezvous point for 45 minutes (using fuel) and then they made 3 attempts at Kokura (using more fuel) that a determination was made if they did not drop the gadget while over Nagasaki they would not have enough fuel to return to any airbase. According to the logs one engine ran out of fuel upon landing in Okinawa and they had 7 gallons of fuel remaining. Had they been carrying the weapon Fat-Man they would not have made it to Okinawa. Again I believe they made the determination to say they had a visual id of the target and dropped the weapon using the radar sighting system. Given the options and the fact that they were witnesses of Hiroshima I am sure they figured what is a few miles when the entire city of Hiroshima was destroyed, so to preserve themselves they dropped it.
I am thinking if one were to check the variance of radar targeting at the time [SHORAN] - they might come up with at least a 2 mile variance.. Just a hunch the meeting notes seem to suggest this as well. [It was agreed that Dr. Stearns and Dr. Dennison should keep themselves continuously informed as to radar developments. If at any time new developments are available which show in combat a marked improvement of accuracy the basic plan may be altered.]