So imagine a world where the nuclear bomb is not possible. World War 2 happens as normal, though likely ending slightly differently. Germany is as it was in 1945. But in Japan things are different. Due to the needs of a land invasion it is likely divided between the USSR and USA much like Korea. Stalin, Atlee and Truman are all in power.
Do you think we could have seen a third world war, fought with conventional forces? My thinking is that as Stalin was incredibly expansionist he may have tried to take more of Europe. Without nuclear weapons in existence a conventional war would seem winnable and he may have been all the more likely to test the US.
In short - yes.
The only reason why there was no WW3 is because the destructive consequences by far outweighed the economic, political and cultural gain from pursuing war from either side. There are number of flashpoints in the cold war that quite clearly would have resulted in war had it not been for the sheer destructive terror of nuclear weapons that made any forms of conventional warfare non beneficial. Just look at the amount of proxy wars that went on for gods sake. There has already been a "third world" war, rather than a "world war three".
This is a very interesting question.
I'd say that yes, there probably would have been. Eisenhower's use of brinkmanship certainly played a factor in the Soviet Union backing down. Without the threat of nuclear weapons, war would have been almost inevitable - both powers actively tried to prevent WW3 because of the absolute destruction that would have come about as a result of the weapons being used.
Keep in mind Eisenhower reduced investment in conventional weapons because he saw nuclear weapons as a sufficient deterrent and "more bang for the buck". Investment would've been different entirely without nuclear weapons, and NATO probably would have been a much more capable conventional fighting force than it was in the Cold War (since it didn't need to be really really strong due to the threat of nuclear weapons).
I'd like to think there wouldn't have been a World War 3. For example, Truman didn't expand the scope of the Korean War in our timeline, but it wasn't due to nuclear weapons. The only opposing force to have nukes were the Soviets, but their delivery systems were pitiful at the time. It was due to the fear of a war with the Soviets in the first place. A WW3 would most likely be suicide on the home front for whichever president got into it.
The destruction caused by World War 2 was something for the Soviets to keep in mind, too - 27 million war dead and a couple of years of Nazi occupation (in the western areas of the SU) speaks for itself. They probably wouldn't be primed for another war by the time Stalin kicked the bucket in 1953, and I don't think Khrushchev would have necessarily a war... though it's extremely difficult to say as brinkmanship dominated foreign affairs, whereas without the threat of nuclear weapons Eisenhower/Dulles might think twice before applying that policy. If they kept brinkmanship, then there'd be a WW3 no doubt.
By the same token, WW3 could've easily resulted out of any of the crises in the Cold War. I'd say most likely over Cuba. If that still happened, the threat of nuclear warfare certainly wouldn't have forced the world leaders to make peace. In that event, there would be a world war. So I'd have to say... yes, there probably would be another world war. There were too many crises that could've easily spiralled out of control without the looming threat of nuclear doom. It is difficult to say, though, as diplomatic interactions and foreign policies would be pretty different without nuclear weapons. There'd be less risks and aggressive moves taken, that's for sure.
Let's put it this way, Operation Unthinkable would likely be taken more seriously, had the Americans not had the nukes to warn off the red army.