For starters, we would not be here. Without nuclear power, thereis no sunlight. A better question would be what if The Manhattan Project had failed. It is understood that the German program was under the impression that the amount of fissionable material would be the size of a beanbag chair. The American project never got rolling until two (German Jewish) little known physicists deduced that the actual size was more like an orange, or maybe a lime. Once this was reviewed and accepted as good science, the push to develop the bomb became, key word here, fundable.
Take away that insight, say into a concentration camp, and all the effort that went into Manhattan goes into conventional war production, and the world is very different after the war.
I must disagree with insanewarrior. It is much easier to build a reactor than to build a bomb. With the machinery of a massive structure available, it is relatively aesy to control the reaction. Stuffing it all into a portable package is the hard part. The economic advantages of fissionable fuel are sufficiently great that nuclear power would occur within a generation at least, say 1960. I can easily see scientific pundits telling students that while nuclear power is feasible, even easy after the basic engineering is done, a nuclear weapon is theoretically impossible.
What would be interesting is the political impact of a Cold War without the threat of the bomb. The Russians became very good with chemical and bioweapons as it is. What if the only thing keeping Stalin out of Berlin in the 1950's was the threat of conventional NATO arms? Would the space race have heated up without the need to build missiles? What impact on the oil situation is nuclear power divorced from military overtones? Would radiation have connotations of cancer treatments?
Interesting.