It seems like there would have been more, for lack of a better term, “modern” execution methods available. Why was hanging chosen?
This one isn't terribly exciting, as even in the recent academic reexamination of the Soviet role at Nuremberg by Francine Hirsh, the story is confirmed by Russian documentation as well as the traditional Western narrative: Russian jurist Iona Nikitchenko - and his bosses all the way up to Stalin - wanted those convicted to hang as common criminals.
French tribunal member Henri Donnedieu de Vabres actually raised the possibility of a firing squad with his fellow jurists before they began deliberating individual cases, but was met with strong arguments against it by Nikitchenko. Unlike many other Soviet objections, on this one United States representative Francis Biddle was in full agreement - a firing squad was not appropriate, the defendants should be hung, and the initial decision was reached without much further debate.
The only brief revisiting of this came when the tribunal debated what to with Jodl and Keitel; de Vabres raised the question again focusing then on if a firing squad might be appropriate for military officers who were following orders, but this time was rebuffed by the British jurist, Sir Geoffery Lawrence, who pointed out the decision had already been made.
The most interesting aspect of this is the timing: as mentioned, this determination took place immediately after the close of the trial, shortly before discussion of the verdicts themselves. As such, the prisoners knew all along that execution was a likely possibility but not precisely how - right until they walked into the docket to hear the verdicts.