It's impossible to answer this with certainty. Significantly, civilian leaders on both sides came to think that any direct confrontation would likely become nuclear.

Robert McNamara, who was Secretary of Defense (1961-1968) under Kennedy and then Johnson, wrote and lectured extensively about this later in life.

For example:

"...[the failed U.S. invasion of] the Bay of Pigs... undoubtedly influenced their thinking, I think that's correct. But, more importantly, from a Cuban and a Russian point of view, they knew what in a sense I really didn't know: we had attempted to assassinate Castro under Eisenhower and under Kennedy and later under Johnson. And in addition to that, major voices in the U.S. were calling for invasion.

"In the first message, Khrushchev said this: 'We and you ought not to pull on the ends of a rope which you have tied the knots of war. Because the more the two of us pull, the tighter the knot will be tied. And then it will be necessary to cut that knot, and what that would mean is not for me to explain to you. I have participated in two wars and know that war ends when it has rolled through cities and villages, everywhere sowing death and destruction. For such is the logic of war. If people do not display wisdom, they will clash like blind moles and then mutual annihilation will commence.'"

Source: The Fog of War: Transcript http://www.errolmorris.com/film/fow_transcript.html

I would argue it would be highly unlikely for there to be a conventional war between NATO and the Warsaw Pact. Any direct conflict between the two would have, most likely, gone nuclear immediately.

As time has worn on Warsaw Pact war plans have begun to become available and, as far as I'm aware, every single one of them calls for the use of nuclear weapons from the start and they are usually measured in the hundreds.

Here is a plan from Czechoslovakia dating from 1964. Whilst a Polish plan from 1970 can be found here. Both employ the judicious use of nuclear weapons from the start of both.

For a decent chunk of the Cold War the Soviet attitude was that nuclear weapons were an extension of conventional weapons and didn't really draw a distinction between the two. Hence these war plans will hundreds of bombs in them.

Eventually they did start to move away from this attitude but on the whole, as far as I'm aware, most Soviet plans, throughout the Cold War, assumed either immediate escalation to the use nuclear weapons or at most a short conventional phase followed by nuclear exchange. (See here for some discussion on Soviet doctrine).

Then it's worth considering that NATO toyed with, on and off, the idea of using their conventional forces solely as a 'tripwire'. The idea being that the conventional forces would buy enough time for NATO to determine that this was the real thing, the Soviets were actually marching on the Rhine, and then NATO would nuke the advancing Soviet forces. (Some discussion of NATO strategy can be found here)

I don't think that ever actually became accepted doctrine (the West Germans were understandably reluctant to sign off on their own voluntary immolation). But more generally there does appear to have been something of an acceptance that, during most of the Cold War, unless NATO got lucky or the Soviets conventional forces were considerably worse than expected, NATO would eventually either have to use nuclear weapons on advancing Soviet troops or concede West Germany.

tl;dr The possibility of any NATO vs Warsaw Pact staying conventional for any significant length of time (and significant in this context is probably no more than a couple of days) is remote in the extreme. Both sides expected to go nuclear early.

The probability of a war in Europe not going nuclear was low. NATO sought to balance the superior numbers of Warsaw Pact conventional forces with its nuclear weapons and in the early days of a Warsaw Pact push west would likely employ tactical nuclear weapons to prevent Soviet gains.

This strategy is why Reagan received significant criticism from the right after the Reykjavik Summit in 1986. Nixon and Kissnger penned an editorial that argued nuclear abolition would encourage Soviet aggression since "the gap in deterrence of conventional attack" would prove too tempting. Brent Scowcroft called the idea an "absolute disaster." John Poindexter, then the National Security Advisor, noted abolition return the US to the "situation which [it] faced in the 1950s" with only a small "chance" of turning back a Soviet assault, and the Joint Chiefs agreed. Based on this it is unlikely that such a conflict would not go nuclear quickly, though there are questions about Reagan's willingness to authorize a nuclear strike.

Whether such a war would involve strategic nuclear weapons aimed at the USSR and USA is a more interesting question, and it is possible to imagine a scenario when nuclear weapons are employed only within Europe, though this is probably not the most likely scenario.