A lot of people who worked on the Manhattan Project (Richard Feynman, for one) justified their participation based on fear that the Nazis would get nuclear weapons first. Was there a Nazi or Japanese equivalent to the Manhattan Project?
Germany was home to the most important scientific developments in early 20th century Physics, so it wouldn't be surprising if the Nazis had a sophisticated program of their own. Then again, a lot of the major scientists were Jewish and fled to the United States before the war started.
Also, when did the Soviets join the race for the bomb? Were they on track to finish before the war ended had the Trinity Test not occurred when it did? Did they start before 1942 when they were still Nazi allies?
On a slightly unrelated note (maybe a bit more speculative), were the Nazis working to get to space? Wernher von Braun, who was arguably the most important single person in NASA history, worked for the Nazi government during WWII. He developed missiles during the war, including the first artificial object to enter space. After it ended he and many other scientists were recruited by the US government and they eventually sent men into space. Did the Nazis have any similar plans, or were they only interested in weapons?
Both the Germans and Japanese technically did have projects devoted to looking into the use of nuclear fission for military purposes, but the programs were mostly stillborn — they were underfunded, exploratory, mostly theoretical programs, not programs like the Manhattan Project, which was developed with the specific purpose of developing a nuclear weapon as soon as possible. There is an important distinction there. As one illustration, the Manhattan Project employed some 600,000 people; the Axis programs did not even employ one tenth of that combined (I doubt they even employed more than 1/100th of that). None of the Axis programs got very far at all by the end of the war.
The German program was under-funded, under-coordinated, and focused almost entirely on reactors from 1942 onward. By the spring of 1945, when the Allies were invading from both sides, they had not yet even gotten a proof-of-concept of a nuclear reactor working (something the US had done in December 1942). The Japanese program was even further away, having concluded that atomic bombs were not something that even the Americans could pull off during World War II. They investigated some isotopic enrichment methods but didn't get very far and never even looked into reactors seriously.
The Germans did pull off some interesting technological developments during the war — most notably the V-2 rocket program — but they did not invest heavily in atomic energy.
The Soviets knew about the American project by 1942 but did not have the resources to invest in a bomb project of their own. From 1942-1945 their atomic project was purely about acquiring espionage information, sorting through and evaluating it, looking into getting uranium resources (the Soviets were uranium-poor at the time, and you need a lot of raw uranium), and planning out a future program. In 1945, Stalin ordered that the project be accelerated into an actual production program. The got their first pilot reactor running by 1946 and tested their first bomb by 1949.
Von Braun and many of the other rocket engineers had gotten into the field because they were interested in space travel. They were inspired by the work of Hermann Oberth, one of the many rocket enthusiasts who recognized the possibility of space travel. The German rocket enthusiasts got hooked up with the Nazis early on and used the military implications as an excuse to get their work funded.
There were no wartime plans to get into space, but it is clear this was one of von Braun's long term goals — for whomever won the war. As you know he was recruited by the USA, ran the Redstone rocket project for the first US ICBMs and satellite rockets, and so on.
If the Nazis had won the war, I am sure they would have also pursued such things — it is in line with their interests, their imaginations. They too were motivated by romantic notions of rocketry (to the point of spending huge amounts of money on them even though they were completely indecisive as military weapons). But it wasn't a wartime goal for obvious reasons.