Unless the Nazi were given a nuke via time travel, there is no way they could of "gotten the bomb" first.
The Nazi military science divisions were competing with each other (even going out of their way to ruin the efforts of each other), fighting over a limited amount of funds. Some of them knew their machines could never work (such as a near orbital bomber that would of had the flying characteristics of a brick in the real world) yet continued to suck up money and resources anyway because they wanted the money/they feared the punishment of failure/they were Allied spies/some combination.
Also, they were not allowed to use any "Jew Science," including Albert Einstein. You kind of need General Relativity to make The Bomb.
So, yeah, time travel is involved.