At peak number of weapons? That wasn't in the 1960s, it was in 1986 (~65,000, excluding those in India, Israel, Pakistan and South Africa)
Peak in the 1960s was in 1967. The Soviets had just under 9,000 warheads, the PRC had ~20, and NATO had 30,000+, virtually all of them being American.
Depending on the scenario, in 1967 the Warsaw Pact and selected allies (chiefly Cuba and the DPRK, with the PRC being possible) are hit by somewhere between 9-10,000 weapons in a full out, no holds barred, exchange. around 2/3 of those are used in counter-force strikes. Every city over 100,000 in the WP is obliterated, Cuba ceases to exist as a civilized state as does the DPRK, and the PRC may catch as many as 500 weapons (the U.S. was not really clear on the fact that the PRC and USSR were not exactly on good terms). Western states are going to absorb around 4,000 weapons, again 2/3 of those, perhaps a bit more, are counter-force. That leaves 1,300 weapons, assume half in Western Europe, the rest in the U.S. With the exception of a few obvious U.S. allies, mainly Australia, Japan and the ROK, the strikes there being at military targets. Australia likely is hit by as few as six warheads.
Around half of the total stockpile for both sides are lost in counter-force attacks, technical failure of either the delivery vehicle or the weapon itself, or are not strategic or tactical long range in nature (torpedo warheads, depth charges, mines, etc). There would also be a small number undergoing maintenance and unavailable.
The Eastern half of Europe, Soviet Asia, and sections of the U.S. are simply uninhabitable due to fallout, frank weapon effects and loss of infrastructure. West Germany is seriously pounded, the UK takes serious damage with major cities wiped out, along with considerable section of the infrastructure, probably enough to cause an utter collapse, France somewhat less than the UK and the rest of NATO being damaged mainly near military bases and strategic assets (oil facilities, major ports, etc.)
The U.S. would actually end the exchange with a useful "2nd strike" or enduring deterrent. The question is if there would be a central government left. Most of the Southern Hemisphere would be more or less untouched (this assumes no use if biological weapons, which, based on the small amount of open source, seems unlikely but is also outside the OP question).
Exchange at max inventory is very different, at least in global impact. There is sufficient overkill built in that strategic assets that either side might be able to take advantage of are going to be hit. This include locations across the Southern Hemisphere, anywhere the USN or Soviet navy could possibly find dock space is gone, every major oil terminal and oil field on the planet is at least targets if not hit. It would be a surprise if any city over 500,000 survives anywhere on Earth. Soviet engineered bio-weapon usage is a virtual certainty both human pathogens and agents designed to destroy crops.
tl;dr: In the 1960s the Southern Hemisphere survives with minor overall damage. In a full 1985 exchange Mad Max is a best case scenario.