If nuclear weapons were never invented, we would be looking at the extension of the the war in the Pacific by at least 9 months possibly a year. Although we had the Japanese completely on the defensive and almost pushed to its limit, we would still have to contend with millions of fiercely loyal Japanese citizens who were fabricating rifles and weapons in their own homes. We would be fighting a mob that was willing to sacrifice themselves for their country.
After the end of World War 2 some time in middle 1946, you would also see multiple changes in military policy, foreign policy, intelligence, etc. all simply because the concept of mutually assured destruction doesn't exist.
In terms of military policy, American and Soviet leaders would be more open to first trike and pre-emptive options, essentially the same exact rules or war that we have seen for centuries. Because of nuclear weapons, neither side wanted to commit to a first strike and risk retaliation from their cold war enemy. Because of this, military leaders were usually hushed by their own leaders.
Foreign policy wouldn't have changed at all since the beginning of the 20th century. Foreign policy changed after WW2, more empahsis on alliances with similar cultures. It is quite possible that organizations such at NATO, the United Nations, Warsaw Pact would have never existed since these organizations were built around nuclear detterence and nucear assistance (i.s. if the Soviet Union launched a nuclear weapon at the United States, the United Kingdom and France would launch back in kind.)
The biggest one, the largest of them all is that our concept of national intelligence, overhead and espionage would not exist. Right after WW2, an Air Force Major petitioned to the Air Force that the United States needed to invest in a military aircraft (U-2 and later, the A-12 (SR-71 military designation)) that would be able to fly at extremely high altitudes to look to see if the Soviet Union was working on nuclear weapons. The military was more interested in nuclear deterrence by developing a betrter first strike weapon, however the CIA was very interested.
Actually, believe it or not but the CIA and KGB did more to prevent nuclear war than any other agency on the planet, both organizations spend billions thwarting the other side and even on a rare occasion, worked together through a third party to prevent WW3. Without the threat of nuclear destruction, no agency would have any incentive to prevent war.
It is my firm believe that the threat of nuclear war that caused the cold war did more for peace in the twentieth century than anything else. Both sides refused to ever fight first and both sides refused to piss the other side off so they simply just had the world's longest staring contest. Without the threat of nuclear weapons being the end game for a conflict, the United States and the Soviet Union would have been more than happy to go to blows, possibly having a WW3 in the early fifties and depending on the out come, a WW4 and WW5. Nuclear weapons kept us from being brash and made us more pragmatic from 1945 to present.