It's worth noting that the Nuremberg trials were not just limited to the International Military Tribunal. That was just the trial of the Nazi leadership, and the tip of the iceberg.

There were other Nuremberg Trials that followed. The big difference between them and the IMT, was that they were entirely under US Jurisdiction and followed American law, compared to the IMT, which, as the name suggests, was "International" in that it had American, British, French and USSR prosecutors and judges, who followed their own standards and procedures when prosecuting the Nazi leadership. Examples would be the NMT (Nuremberg military tribunal) Einsatzgruppen trial and the Nuremberg Doctor's trial.

Looking at the other Nuremberg trials, it becomes clear that the majority of people on trial did in fact survive. This was due to a sharp change in Allied, specifically American policy. In the first place, the Nuremberg Trials never enjoyed popular support. The main driving force was US Supreme court Justice and Prosecutor Robert H. Jackson, but that was it. The American and British Leadership were more concerned about the growing threat of Soviet Communism than past Nazi war crimes, and acted accordingly. The researcher Ulf Schmidt contrasted Operation Paperclip, the program to recruit Nazi scientists and others, with Allied, and specifically American prosecution of war Criminals. Operation Paperclip was meticulously planned and involved the highest levels of the US Government and Military. 3,000 T force members were involved in its planning and execution. In contrast, even with Jackson's determination, the US lacked a consistent approach or plan in prosecuting Nazis, and the investigators like Leo Alexander or Benjamin Ferencz had to go their own way. Eventually, after the Nuremberg IMT even Jackson gave up due to lack of support from his superiors.

The end result was that very few people were actually convicted. For example, Benjamin Ferencz mentioned that out of 3,000 Einsatzgruppen, he was only able to bring 20-30 to trial, due to the vast majority "slipping through the cracks". They would not be tried until the Ulm Einsatzgruppen trial in the late 50s, which only happened by accident, due to a former Einsatzgruppen member feeling confident and thinking he could claim his pensions, due to the large lack of interest the Germans and US had in prosecuting Nazi crimes.

Another example was the Nuremberg Doctor's trial. 23 people were tried. 7 were acquitted (with one, Kurt Blome, being subsequently recruited by the US), 7 were sentenced to death, and the remaining 9 were given life sentences, which were later commuted to timely sentences or time served. In other words, the majority of defendants were treated lightly and allowed to go free.

There is some flexibility in the answer depending on how you define "leader", but by most definitions, yes, some Nazi leaders survived. The most notable were Karl Dönitz, Albert Speer, and Rudolph Hess.

Dönitz was the commander of the U-boat force, then commander of the navy from 1943, and then Hitler's successor after Hitler's suicide. He was very Nazi, an admirer of Hitler, and anti-Semite and anti-communistic. He worked to spread Nazism in the navy. He claimed (probably falsely, but this could not be proven) that he knew nothing of the Holocaust. He was sentenced to 10 years. After he was released from prison at the age of 65, he lived a quiet retirement, writing two books, one about his Nazi years and the war, and one about his earlier life. His memoirs presented himself as a soldier, not a politician, and therefore not responsible for Nazi atrocities - a common theme in many Nazi memoirs, and at least a bending of the truth, if not an outright shattering of it.

The judgment against Dönitz: https://avalon.law.yale.edu/imt/juddoeni.asp

Unlike Dönitz, Speer was very much a politician and not a soldier. Speer joined the Nazi party in 1931, and shortly after the Nazi takeover of power, he was in charge of the Chief Office for Construction - basically, Hitler's architect. During the war, he moved into the Nazi inner circle, as Minister of Armaments. His 20 year sentence was for his knowing use of slave labour in this role. He had not been involved in the planning for the war, and he was not found guilty of "crimes against peace" (if he had, he would probably have been executed). Speer was 61 when he was released in 1966. He tried to work as an architect, but was unsuccessful. He became a celebrity (anti-celebrity?) with the release of his memoirs (the English title was Inside the Third Reich), which provided an inner view of much of the working of the Nazi government, while presenting himself as "only following order" and not knowing about the Holocaust. His memoirs portray himself as a "good Nazi", a patriot rather than a criminal. However, his memoirs were even less honest than Dönitz's - not only was Speer deeply criminal, he was also a lying sack of shit. Not only did he lie to his post-war readers, he also lied to the German public during the war, using selected statistics as propaganda claims of huge increased in armaments and munitions production.

The judgment against Speer: https://avalon.law.yale.edu/imt/judspeer.asp

Hess became a Hitler loyalist as early as 1920, and, imprisoned with Hitler, was one of the people who Hitler dictated Mein Kampf to. Unlike other Nazi leaders, stayed a Hitler loyalist rather than building his own little empire in the Nazi apparatus. Hess was Hitler's deputy, Hitler's secretary (until replaced by Martin Bormann in 1939), and Minister without Portfolio from 1933. In 1939, he was named Hitler's successor's successor, next in line to the succession to the leadership of Germany, after Hermann Göring. In 1941, he flew to the UK on an unofficial and insane mission to seek peace with the UK. He was captured. He was tried at Nuremberg for his part in planning and starting the war, and sentenced to life imprisonment.

The judgment against Hess: https://avalon.law.yale.edu/imt/judhess.asp

There were questions about his sanity, but the conclusion was:

That Hess acts in an abnormal manner, suffers from loss of memory, and has mentally deteriorated during this trial, may be true. But there is nothing to show that he does not realise the nature of the charges against him, or is incapable of defending himself. He was ably represented at the trial by counsel, appointed for that purpose by the Tribunal. There is no suggestion that Hess was not completely sane when the acts charged against him were committed.

Other Nazi leaders survived, too:

• Walther Funk (Minister of Economics 1938-1945, President of the Reichsbank 1939-1945). Deeply involved in the Holocaust, in the role of confiscation and sale of the property of Jews. Sentenced to life imprisonment, and released in 1957 due to ill health. He died of diabetes 3 years later. https://avalon.law.yale.edu/imt/judfunk.asp

• Baron Konstantin von Neurath (Foreign Minister 1932-1938 (replaced by Ribbentrop) and Reichsprotektor of Bohemia and Moravia 1939-1943 (with real power transferred to Reinhard Heydrich in 1941, because von Neurath was seen as too lenient). Sentenced to 15 years, and released early in 1954 due to ill health following a heart attack, and died 2 years later. https://avalon.law.yale.edu/imt/judneur.asp

• Baldur von Schirach (head of the Hitler Youth 1933-1940, Gauleiter of Vienna 1940-1945). In the latter role, he was responsible for sending about 60,000 Jews to concentration camps. He claimed to know nothing about the Holocaust. He served his full 20 year sentence, and was 59 years old when released, living a quiet retirement and writing his memoirs (low-profile compared to Speer and Dönitz). https://avalon.law.yale.edu/imt/judschir.asp

Beyond these individuals, there were others charged with war crimes and not executed (either acquitted or sentenced to imprisonment), who were involved in the running of the Nazi state, and some might consider them "leaders", and others would not. The above are those who held ministerial level posts in the government, or higher, and von Schirach who was head of the Hitler Youth. Von Schirach was also a Gauleiter; of Gauleiters who survived to the end of the war, eight were executed, and 20 were imprisoned. Some others were not charged and tried for war crimes, and remained free. For a convenient list of Gauleiters and their fates, see https://en.wikipedia.org/wiki/List_of_Gauleiters