- The Nazi party wouldn't have been as influential as it was in its early years without Hitler. It was started by an unimaginative nobody named Anton Drexler as the German Workers' Party, and it wasn't particularly influential. Hitler changed that; his speeches were what made the party popular. Emboldened, he completely changed the party's focus; he basically rewrote their manifesto to solidify the party's platform, added the "National Socialist" moniker to better appeal to nationalist sentiment, and pushed Drexler aside into a life of political obscurity. He didn't plan the 1923 Beer Hall Putsch — other far-right figures had beaten him to the punch — but when those figures decided to abandon it, Hitler forced them to go along with it (at gunpoint, no less). It failed miserably, but it did make Hitler and his movement famous across Germany. Even after a brief jail term and a lengthy absence from politics, Hitler had already built a reputation for himself, so when he came back, he was easily able to take the Party back from the moderates who ran it in his absence.
Furthermore, after Hitler came back, he ensured that the "National Socialist" bit didn't actually mean "socialist". Many existing Party members, like Gregor Strasser, were more open to friendly relations with the Soviet Union and some level of co-existence with the German Communists. Hitler was dead-set on crushing Bolshevism and purged those left-leaning elements from his party. In fact, he even ordered Strasser to refuse an appointment to be Vice-Chancellor.
- Many other candidates to start the war instead of Hitler only came to power through Hitler's help. It's sometimes claimed that another notorious Nazi figure, such as Heinrich Himmler or Joseph Göbbels, would have taken Hitler's place if a time traveler eliminated Hitler. But without Hitler, those people would have been on the fringes of the Party. Only after Hitler radicalized the Party would their ideas become valuable; he essentially brought them with him to positions of power. And none of them had the charisma or reputation Hitler himself had built to be a plausible populist leader. What they did have, though, were the smarts to help build that reputation into a real personality cult. This made the Nazis much more of a totalitarian group than they otherwise would have been.
- Hitler was the most prominent mind behind German expansionism and The Holocaust. This isn't to say that racism and anti-semitism weren't a thing in Germany before he came to power, but Hitler was the one who wanted to make it key to the party's platform. He bolstered this sentiment by tying it in with populism and the German national identity. He made several pushes to recover or gain German territory that saner politicians and military minds would have considered too risky. He was not at all bothered by the prospect of war. And we know that he did personally order the Holocaust, even if other Nazis carried out those orders just as enthusiastically. There may have been war and anti-semitism without Hitler, but likely not on the scale of what actually happened.
- The economic and political meltdown of the Weimar Republic would not have been enough to spark the war on its own. One common thread is that Hitler simply took advantage of the massive increase in political extremism seen as a response to Germany's dire situation in the 1930s. This, however, does not mean that the type of extremism would have been the same as what we actually got. Many of these groups were hopelessly fragmented, and there was a schism between leftists who blamed extreme capitalism for the problem and right-wingers who blamed it on non-German influences. Hitler was a charismatic right-winger who converted many more leftist German extremists to his side. As it happened, Joseph Stalin ordered the German Communists not to oppose Hitler or ally with more moderate socialists, thinking that Hitler could be useful to him — without that, it's possible that a leftist extremist group would have emerged in control of Germany. Even then, they could easily have fallen into a Civil War with the right-wing remnants, which would be pretty bad — after all, the Soviet Union would probably have gotten involved — but (a) not the same as what we got, and (b) likely not as bad because nobody would have been ready for civil war to break out in Germany.
- Hitler's aggressive foreign policy and swift invasions influenced other actors during the war. It's likely that his actions either emboldened or scared other countries into being more aggressive themselves. Fascist Italy might not have invaded Abyssinia if Hitler's success wasn't making Mussolini look bad. Japan might not have been so desperate as to attack Pearl Harbor (although they were probably invading China one way or another). A more moderate Germany might not have propped up fascist dictators elsewhere, like Francisco Franco in Spain (meaning the communists could have won the Spanish Civil War).
- The Soviet Union was not likely to take Germany's place as the bad guys. One thread to this trope has the Nazis never take power, but for an emboldened Joseph Stalin to try and take over Europe and thus become the main antagonist instead of Hitler. But this is unlikely for several reasons. First, their military was woefully incompetent after Stalin's purges at the time, so they wouldn't have been as much of a initial threat. Second, they were extremely cautious about uniting the world against them, Stalin's preference was instead for the capitalists fight each other rather than get involved himself. (Indeed, the Soviet Union originally intended the Axis Alliance as a way to do this, but the Nazis took only six weeks to conquer Europe before turning toward Russia.) While the Soviet Union ultimately would have become stronger in the 1940s without getting invaded by the Nazis, it is likely not going to be enough to win a war with the entire war and this is something they would be acutely aware of.
And they wouldn't have been close with Japan, either. Japan was actually terrified of the Soviet Union during the war. They might have conclusively beaten Russia in 1905, but that was in a previous era. Most recently, various limited border wars in Mongolia in the 1930's had completely reversed this. A then-obscure Russian general called Zhukov had hammered them with better men, superior generalship and tanks which vastly outperformed anything Japan could hope to field. After this, Japan even kept a large contingent of troops on the Chinese-Russian border throughout the war and very carefully refrained from doing anything that might provoke their neighbor.note When the Soviets attacked in August 1945, they rolled over the Japanese forces in Manchuria in six weeks.
- No, nor would Fascist Italy. Italy did get on the fascism train before anyone else and had conquered Ethiopia by 1937. But that took a protracted and costly war, which would have left them ill-equipped to take over the rest of Europe — which they proved when Hitler roped them into participating in the Spanish Civil War. And they still needed the military to hold on to their North African claims and put down the remaining pockets of La Résistance. Emperor Haile Selassie of Ethiopia was still campaigning passionately for his nation's freedom and the downfall of fascism, despite being in exile — he gave a stirring speech to the League of Nations trying to head off the world's Moral Myopia, saying, "It is us today. It will be you tomorrow." All of this led to an eventual collapse of public support for the fledgling Italo-African Empire, and outraged Italians even turned on their own military, eventually leading to Mussolini's ugly demise.
That said, Italy might have been the best-equipped to replace Nazi Germany as the Big Bad of World War II. Italy had already survived embargoes and international pressure in response to their Ethiopian conquest, and Hitler allied with them thinking they were the role model for European fascism, not the other way around. And Mussolini was a clever man who could lead this empire — as a former journalist, he had a keen gauge for public opinion, and as a former Cold Sniper, he knew the value of patient and methodical warfare. His main failure in "our" timeline was allowing Hitler to goad him into more expensive wars to protect his pride. Even still, if Mussolini wanted to fight a war as bad as "our" World War II, he would have to have been as aggressive as Hitler, and we know that his aggression in "our" timeline led to the Allies completely kicking his ass in North Africa, so what would have resulted from a Mussolini-led Europe might have been bad, but it wouldn't have been World War II.
Analysis / Hitler's Time-Travel Exemption Act
One way for Hitler's Time-Travel Exemption Act to work would be to claim that Hitler himself wasn't necessary for Germany to turn into the Third Reich and kickstart World War II. The idea here is that removing Hitler at any point in his timeline — even before he was even born — wouldn't prevent the war from happening and shaking the world at least as badly as it did in "our" timeline. However, this would likely not be the case for the following reasons: