How was the POTUS involved? What did he have to sign? Which button did he have to push?
Follow-up: How would the process go today, in case for example the US decided to launch a nuclear warhead? (This is of course hypothetical but I would love to know the actual system)
The president authorised the programme as a whole (Roosevelt) and approved the use of the weapons on Japan along with a set of guidelines on when to drop them and a list of potential targets (Truman). The decision on precisely when and precisely where was left to the discretion of the people on the ground in Asia based on the weather or other conditions. If you'd like to see the actual document approved by the President, this would be the last point of contact for Truman, dated 25 July 1945, and can be found here.
Had there been good weather on August 3, there would have been a much larger delay between the two bombs being dropped. However such decisions were not being personally micromanaged by Truman.
The President did not have a large red button to push, and indeed the first bombs did not have the kind of extreme safety measures attached that we imagine today when we think of launch keys and PAL codes.
I do not have detailed information about the current process for nuclear authorisation in the US, sorry.
I will ping /u/restricteddata as he is unequivocally the expert above all others on here in this.
However when it comes to Truman in WW2 you should not think of it as he took an affirmative decision, "the plane is in the air Mr. President, Do we Drop?"
It was basically never even considered that they wouldnt be used, all this time, money, and manpower had gone into a new weapon, it would have been a colossal waste to not do anything with them.
And then after Hiroshima, since the operational plan gave the forward commanders at Tinian,including Tibbets, the authority to change the date of the 2nd mission for weather concerns that is exactly what they did, greatly shrinking the time between bombs without any order by the President. So Truman was distressingly out of the loop, or uninterested with regards to many aspects of the bombs, and only after their effects became clear through his possible misconceptions, and their power was clear did he order no further release without his say so.
If I may piggy back on your question: Post war, what did the decision making process look like for tactical nuclear weapons like the Davy Crockett that were deployed in areas like the Fulda Gap? Would POTUS even be involved in the decision loop?
During what time period?