How apocalyptic would a full scale (if such a scale was a reasonable expectation) nuclear exchange actually have been during the Cold War? Did the militaries of the powers assess the probable result accurately?

Very apocalyptic. During the height of the Cold War the US and USSR each had around ten thousand nuclear weapons in service, ready to be used. They varied in yield from around 100 kilotons to several megatons, but most of them could have been deployed within a matter of minutes to hours.

One thing to keep in mind about nuclear weapon strikes is that modern cities are generally much more dense than Hiroshima and Nagasaki were in 1945. Coupled with the increased yield of modern warheads that translates to a much greater death toll, as well as a much greater level of destruction.

A single 100Kt bomb would cause massive destruction and a significant (>50%) loss of life over an area of about 14 km^2. Multiply that by 10,000 warheads and you get 140,000 km^2 of destruction (per arsenal). This is a good rough estimate but it shorts the total megatonnage of either the US or USSR's arsenals from the 1960s through the '80s by around a factor of 4 or so. Most urban areas have a density of between 2,000 to 10,000+ people per km^2 so if all of that destruction was focused on population centers then the death toll would easily be hundreds of millions per side, potentially up to billions depending on targeting.

Moreover, the release of roughly 8 million kilotons worth of radioactive fallout would have disastrous consequences for the survivors, killing millions downwind of the targeted cities. Meanwhile, at each of the cities hit by strikes there would be enormous fires which would likely rage out of control, causing further destruction and loss of life on a scale that would be shocking and appalling in its own right in any other context.

Potentially, though research is inconclusive, the debris and smoke from the strikes and their aftermath could cause a worldwide "nuclear winter", which would put a serious hitch in agricultural output for at least several years, perhaps longer.

And, of course, the targets would be chosen for maximum impact. Centers of government, centers of commerce and trade (major shipping ports), centers of industry, military control facilities, and population centers. The vast destruction would cause life as we know it to come to an immediate stop. Government and commerce would come to an end. Chaos would ensue as people tried to find safety in the immediate aftermath (fleeing from fires, radiation, and possibly anarchy or ongoing conventional warfare). Even if everyone remained civil and organized it would still take considerable time to get back to the business of living, the shocks to commerce, trade, and agriculture would have a high chance of leaving a large number of people with insufficient access to enough food to survive. It would take years at a minimum for the world to settle back into some semblance of normalcy.