Between 1960-1985, can we have a World War Three that does not involve the US, USSR, China, or Great Britain? Bonus points if it is between unaligned countries and involves three or more theaters of war.
It's hard to call something a world war if you start removing all the big countries. But given your list, it have to be colonial (so does decolonisation count?)
The Second Congo War was called the African world war. It was pretty big and bloody. The Rwandan genocide may be the trigger. Have it pushed back a few decades to comply with your deadline would go halfway.
Now, depending on exactly when it happens, you can have a South African, Portugese and Cuban interference. And I suppose some other South American players can be proxies for the US. France may intervene.