Let's say that Hitler started working on the bomb early, say in late 1939. Lack of industrial resources, lack of sufficient skilled labor (they'd mostly end up in the army on the Eastern Front), lack of top-flight scientists (they'd either fled or were in camps, or were of dubious loyalty such as Heisenberg) would make it uncertain the Germans could get the bomb before the Americans did. If they really put huge resources into it, that would merely mean the Americans would put even huger resources than in OTL into the Manhattan Project and still get there first. Plus the British would upgrade their ability to deliver biological and chemical weapons.
And let's say Hitler was able to produce a few planes that could deliver the bomb--he still has to deal with the formidable air defenses of the UK and of the USSR (the latter by 1944 had air supremacy over the Eastern front). And Hitler would have to choose whether to use it over London or over the greater threat of the Soviet Union. The Soviet front lines were so wide that any bomb there would be a drop in the bucket. If he drops it on London, the British (extrapolating from the pace of the American program) would know he probably didn't have a second bomb ready and certainly not a third, and they would hit the Germans with enough anthrax to make the country unlivable for decades plus start all-out use of chemical weapons. If Hitler dropped the bomb on Moscow (assuming he could get it there), the surviving Soviet leaders would know (from their spies in the U.S.) that Hitler probably didn't have more bombs and thus they would continue their offensive.
Knowing the Soviets, the result of Hitler nuking their country would be that the Soviet troops once they got into Germany would kill millions and turn East Germany into a Soviet territory peopled by Russians. No East Germany as in OTL; in fact, no East Germans at all.
The only other alternative for Hitler is to nuke on D-Day; but he could only effectively hit one beach. Bad, but not necessarily fatal for the invasion. America would go wild, and strain its resources to the limit to retaliate. Germany would be bombed beyond the Stone Age into Jurassic Park Age.