Actually, Yes, but the Hiroshima event wouldn't have qualified. The 25th Amendment allows for procedures for the President to declare himself temperarily unable to discharge the duties of the office by a letter to both the Speaker of the House and President Pro Tempore of the Senate. In such an event the Vice President becomes the Acting President and has all the duties of the President while POTUS is relieved. There has to date been three terms of an Acting President in U.S. History: then Vice President George HW Bush, while serving as VPOTUS to Ronald Regan (July 13, 1985 from 1128 to 1922 EST), and Dick Cheney in two separate incidents under George W Bush(June 29, 2002, from 0709 to 0924 EST and again on July 21, 2007, from 0709 to 0924 EST.). In all cases, the President was scheuled for surgery that required his anesthetization to perform.
While the Constitution doesn't say it specifically, the reason for this part of the 25th Amendment is specifically for the transfer of the Football to the Vice President. By law, only the President may order a nuclear strike (The Secratary of Defense can confirm the order is from the President... he can't overrule the President once the order is made). This presented a problem in the situation where the President was aware he couldn't access the Football such as a surgery, because the Soviets (or anyone but lets be real about who was the bad guy during the era this Amendment was passed) would be able to launch a First Strike assault without fear of a Second Strike retaliation anytime the President was Scheduled for surgery... which would be public knowledge. As bad as the onset of Nuclear War would be, it was deemed that stopping a doctor who is elbow deep in Leader of the Free World to open the Football was not an ideal situation. So this specific mechanism was introduced in law to allow the President to temporarily step down and empower the Vice President to act in his stead... up to and including deploying nuclear strikes.
Similarly, the "Designated Survivor" in the Cabinet will also have a Football present during the event where they are designated the "Designated Survivor". This is different however in that the Designated Survivor is only named during major events when the entire Presidential Line of Sucession is present in a single event (this happens at least once a year, during the State of the Union) and is usually a low on the totem pole member of the succession line. Ordinarily, the Cabinet is not followed with the Football on a day to day basis.
While we're on the subject, Operation LOOKINGGLASS, which are a series of Planes that are purpose built for Nuclear Command and Control personnel and would be the aircraft of first choice to evacuate the President as opposed to the tradditional Air Force One planes under the best case scenario (The Air Force One planes can do in a pinch, but LOOKINGLASS is preferred). To prevent any foreign agents from getting the wrong idea and accidently starting a Nuclear War, the President is never allowed to fly in these aircraft unless its "Not a Drill". So during Drills, the part of the President will be played by a government worker who is definitely not the President, but will be treated as if he was for the duration of the drill. Normally his "acting" is scripted so any major decision is determined by the people writing the particular War Game scenario. Similarly, War Games will never use "Defcon" designations with a series of Code Words substitute in that are used in place for training only. For example, the highest DEFCON, DEFCON 1 is (in what has to be some black humor) COCKED PISTOL in any War Game scenario. And lest you think this is overly paranoid, an accidental Nuclear War was nearly started in 1983 exercise Able Archer 83 specifically because the Soviet Union thought that the training terms were just codes for the Real Defcon (at the time, they were under the impression that the U.S. would use a war game as cover for mobilization for a real attack. And Able Archer was designed to simulate an escalation to Nuclear War with the Soviets (specifically, the Soviets launched the First Strike in the scenario). To Soviet observers, it was way to close to what they thought the war would look like.). It wasn't helped that they almost started a nuclear war by accident months earlier... which was only stopped because a colonel didn't think it looked right... so they were on edge to begin with.).