If a non-nuclear WWIII happened in either the 1960s, 1970s or 1980s (more specifically 1983 due to Able Archer 83) I can only imagine that it would be essentially WWII all over again (in terms of casualties w/o nuclear bombs and overall scale of the conflict) except this time instead of Allies (Democracy) vs Axis (Fascism) (like in the previous conflict) it would be former allies (USA and USSR) during it out with having their own military alliances NATO (Capitalism) and Warsaw Pact (Communism) respectively. Germany would essentially be the main battleground for World War 3 as it is divided between East and West Germany and American/French/British/West German troops and Soviet/East German troops stationed there would be invading each other for full control of the country as having Germany reunified under one government by force would be a strategic advantage, whoever controls Germany well they control the gateway to the enemy's territory. You can also imagine the conflict spreading to Asia as well after all North Korea still did have ties to the Soviet Union and South Korea/Japan are also allied with the United States so when WW3 breaks out well expect North Korea to invade and try to reunify South Korea under their banner with some support from the Soviets while the American troops stationed in South Korea would counter respond. Then we have Greece and Turkey there the easternmost NATO members bordering hostile Warsaw Pact neighbors and since World War 3 does at least break out in this scenario well the USSR would invade Turkey and Greece while Bulgaria invades Greece too. Israel would also join the NATO/U.S. side since well you know. Then we have the issue of neutral nations such as Yugoslavia, Albania, Switzerland, Iraq and China despite being both communist they didn't side with the Warsaw Pact with at least 3 of the 5 nations I've mentioned having broke off ties with the Soviet Union and for the Swiss they're already neutral when WW3 breaks out I don't see them siding with either side they might even become safe havens for any war refugees from the Eastern and Western Blocs respectively. We also have other Communist Asian states such as Vietnam, Laos, and Cambodia they could also join the side of the Soviet Union and fight the United States given all 3 countries history with the Vietnam War but the U.S. might also get Malaysia and Thailand on their side to counter the Vietnamese, Lao, and Cambodians. Cuba would also side with the Soviets given previous hostility towards the United States. I pretty much see World War 3 fought mostly in Europe and Asia with occasional exceptions such as Israel and Cuba.