Plans over what to do with Nazi leadership were debated between the Allied leadership and shifted throughout the course of the war. I don’t believe its truly possible to definitively state that any single course of action would have been planned and followed through had Hitler been taken alive. While the Moscow Declaration of 1943 did state that punishment for crimes with no specific geographic locale would be handled jointly by the Allied powers, this could be open to interpretation. As the Red Army reached Berlin significantly before the other Allied powers, if Hitler we’re alive he would have surely been under Soviet custody. Whether Stalin would claim that the extreme Soviet losses in the course of the war gave a specific geographic locale for (many of) Hitler’s crimes and thus allowed themselves to take charge in meting out punishment is a highly likely possibility, it cannot be positively concluded.
As for what the specific punishment would have been, as I previously mentioned the thoughts of the Allied leadership on the matter of general post-victory punishment shifted as the war progressed and concluded. There was disagreement over how to punish Nazis generally upon the war’s conclusion at the Tehran Conference in 1943. The intent of Stalin (who would have had custody of Hitler) at the conference can be seen in his suggestion to simply summarily execute roughly 50,000 German officers. A simple summary execution (whether under official Red Army auspices or more akin to the fate of Muammar Gaddafi), is certainly one likely possibility. Although the other Allied leaders were somewhat less bloodthirsty than Stalin in their thinking regarding the punishment of Nazi leadership, immediately executing Hitler and definitely ending the war in Europe is a likely scenario.
On the note of trials, it is highly unlikely that Hitler would have been prosecuted by the German people. The judiciary of Germany was comprised just about exclusively of Nazi party members. The only Germans who could even theoretically prosecute Hitler somewhat reliably would have been those with communist leanings, which simply puts the ball into Stalin’s court regardless. Were a trial to be held under strictly Soviet custody, there is absolutely no doubt that the event would be a show trial in Moscow leading to inevitable execution. If Stalin were instead to abide by the previous arrangement for joint decision making on the punishment of Nazi leadership, Hitler would have been tried jointly by the Allies at Nuremburg or potentially separately at a unique trial for himself alone. It should be noted that, essentially, all roads lead to execution. Any plans for the Denazification of Germany would have been seriously hampered by Hitler still being alive. The risk of Hitler returning, being freed, or acting as a living figurehead for a potential resurgence of Nazism would simply be too great. The only real questions are the exact method leading up to execution, and the extent of involvement from Truman and Churchill in it.
[The Post-war plans for the Nazi leadership in general differed between the Allied leadership. Given that Hitler was not captured alive, it cannot be definitively said that the potential road map for Allied leadership jointly dealing with Nazi leadership would have even been followed in Hitler’s case. The only thing for certain is the conclusion: inevitable execution.]