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.