This is a really good question because most people forget that the NUMBER ONE American policy objective in Vietnam was DON'T LET THIS ESCALATE INTO WW3. That was our absolute #1 goal. Actually winning the war was #2, maybe #3. This explains a lot when you think about it.
If you had had a conventional NATO/ Warsaw Pact war in the late 60s, it would have been very, very messy. Europe would have been burned to the ground for the third time in 50 years. You would have seen the WP forces push west, and NATO would have done their best to stop them short of the Rhine, or failing that the Pyrenees and the English Channel.
The odds of this staying non-nuclear are rather low. You might want to google up "OPERATION DROPSHOT" and "SEVEN DAYS TO THE RIVER RHINE" to see what the actual war plans called for.
MUCH would have relied on how the diplomatic game played out. It is not at all impossible that several NATO members would have said "Hell, I didn't sign up for this!" and just ignored their treaty commitments and declared neutrality. This would have been easier for some nations than others. Denmark, The Netherlands, Belgium, and West Germany would have had Soviet troops physically on their soil, so absent an actual surrender, opting out of the war would have been difficult for them. Italy, France, Spain, Greece, Portugal and to an extent Turkey and even Canada would have had an easier time backing out, depending on what their government thought was best.
In the worst case scenario with the Russians reaching the Rhine and the French either declaring neutrality or surrendering, the UK would have staged a second Dunkirk, pulled back across the channel, and replayed The Battle of Britain. If they would stay in the war at that point would depend on what their then current leadership wanted.
CENTO and SEATO still existed at this time so what their members would have done is also a serious question. The Warsaw Pact governments would have all been onboard with the war, but there would have very likely been passive resistance from the people of Eastern Europe, if not outright rioting.
India and Pakistan would have, no doubt, used the outbreak of war as a pretext to attack each other (with India almost certainly being on the Soviet side). North Korea would have attacked South Korea as well (if only to keep the U.S. forces there occupied). The Arabs were always looking for a good excuse to kill Israelis back then, and with the US unable to send military aid and Israel not having the bomb yet, they would have undoubtedly taken the opportunity to attack Israel, and quite likely would wiped Israel off the map. Not without cost, but they would have been happy to take ANY sort of losses if it meant wiping out Israel. At the time Iran was still a staunch US ally, and Egypt a Soviet one, so things would have been very... different... in the Middle East.
Castro would have made sure Cuba was in the war. IIRC his eagerness to nuke the US during the Cuban Missile Crisis terrified even the Soviets, so there would have been shooting at GITMO, if nowhere else. If Castro would have been deposed by slightly saner leadership or not is an open question. Given it's location the U.S. would have had to neutralize Cuba as a threat early on, though exactly what form that "neutralization" would have taken would have depended on what sort of resources the USA had available at the time,
The BIG wild card is what China would have done. IRL China and the USSR had a seven month undeclared shooting match on their mutual border in late 1969, just as India and China did in 1962. They might have stayed out of the fight, joined "The Great Socialist War of Liberation" against the U.S., or (and this is the least likely outcome) allied with the U.S. That last one was referred to as "The Soviet Nightmare Scenario"... a Chinese Army with an almost infinite supply of men, equipped by the USA with an almost infinite supply of gear. It was unlikely to occur in the 1980s, and VERY unlikely to occur in the 1960s, but it still kept the Soviet general staff up at night.
In an NON-nuclear situation eventually the U.S. economic superiority would have won out. How this would have been expressed and what form it would have taken we can't know; but unless the Soviets found some way of destroying the American industrial base, sooner or later they would have found themselves facing more tanks, ships, and planes than they had the ability to stop.
Furthermore (as the Ukraine invasion again showed) logistics has never been Russian/Soviet strong suit. The Soviet economy might have been good at turning out tanks, but it sucked at turning out the trucks needed to keep those tanks supplied with gasoline, and the tank crews supplied with ammo, food, warm clothing... and it also sucked at producing food, warm clothing and gasoline for that matter...not to mention maps, radios, spare parts and medical supplies and pretty much everything else. Stealing trucks and trains from the civilian economy would have made the already FUBAR Soviet food distribution system even worse, and something akin to the food riots that took Germany out of WW1 would have been a very real possibility. The Soviets would have STARTED the war with an economy that was unable to properly feed and clothe all its' people and things would have just gotten worse from there... and that doesn't account for various "liberation movements" springing up in Eastern Europe and blowing up trains and bridges and such. Things would have been cold, hungry, scary and dark in Europe.
Again, the odds of this thing staying non-nuclear would have been very, very low. NATO relied on using tactical nukes to stop the Soviets east of the Rhine, and the Soviets planned to use tactical nukes to get to the Rhine, not to mention the number of nuclear torpedoes that were loaded on submarines and air craft carriers at that point. When nukes started flying the most likely outcome would have been something akin to the book "When Angels Wept" (which is an alternate history where the Cuban Missile crisis "goes hot" and the nukes start flying). Europe would have been thoroughly thrashed for the third time in 50 years. The U.S. would have taken massive losses but still survived with tens of millions dead, a trashed out infrastructure, and several urban/industrial areas just ... gone...(something akin to what happened to the USSR in WW2, but on a slightly larger scale). Recovery (to the extent it was possible) would have taken several decades, at the very least. The USSR would have been entirely obliterated. American nuclear superiority was a lot larger than either one of the superpowers knew at the time, and you would have seen a lot of situations where U.S. nukes simply "made the rubble bounce". Much of Russia would likely have remained uninhabited, or at least uncontacted, till the early 21st Century. China would have probably caught it's fair (or unfair) share of the American nuclear arsenal as well... but the casualties (at least in human terms) wouldn't have been much worse than what they suffered in the collectivization famines, the Great Leap Forward, and the Cultural Revolution anyway... which is fairly ironic. Anyone with the money and ability to do so would have moved to Australia, New Zeeland, South Africa, Argentina or Brazil.