The Soviet Union was never officially a combatant in the conflict, but did have strong ties to the North, and to China. With the Soviet Nuclear program successfully developed by 1949 did any parties to the conflict consider nuclear war a possibility or likelihood?
The Soviet arsenal during the period of the war was understood to be small and very limited in range. They had only detonated a single prototype in 1949, and even the most paranoid fears did not imagine they would have anything like the capability to really develop hundreds or thousands of weapons instantly. They also had very limited ability to deliver weapons to their targets: they were reliant on bombers, which could be detected and shot down, and only had limited range. And the bombs that were usable during the Korean War were of relatively limited explosive power — more comparable to the weapons of World War II than the weapons that would be usable by the mid- and late-1950s (hydrogen bombs, etc.).
So while the idea of the Soviets having a bomb or two that could be deployed to some target near the USSR (either in Asia or Western Europe) was not totally out of the picture, the idea that they could escalate into what we think of as a full-nuclear exchange was something still for the future (the US estimated this would not really be a problem until the later 1950s). On the other hand, the US had an arsenal that was in the hundreds of weapons and growing, and was beginning to pursue the strategy of "basing" nuclear weapons and bombers abroad, so that they would have the ability to really threaten to destroy the USSR.
The main "escalation" fear of the Korean War was not that a nuclear war would start, but that the Chinese would get involved in a much larger way than they already were. If that happened, it would possibly cease to be a relatively localized war and become something larger and more sprawling, something that could potentially pull the Soviets in as well, which could commit the US to a much more unpleasant sort of military endeavor that could range from Asia to Europe. In such a situation, nuclear weapons could play a role, but it isn't the "end of the world in X hours" scenario that one would later imagine it to become, it was more like "World War II but with (more) nukes," which is still a pretty ugly thing to contemplate.
So the answer is, sure, nuclear weapons use was part of the thinking, but it wasn't probably how you would imagine "nuclear war" to look like even by the late 1950s, much less the 1960s (when long-range nuclear-armed missiles began to be deployed) or beyond.
