What would happen if the USSR went to war with the US and its allies...?

Had a WWIII involved a nuclear exchange, both sides would've been destroyed along with civilization generally.

But what if the fighting were kept conventional? (This wasn't impossible since neither side used poison gas in WWII though both sides had it, and everyone recognized that nuclear war was suicidal.) In a purely conventional WWIII the USSR or Warsaw Pact would've won. The USSR had the advantages of dictatorship, enabling it to build much bigger conventional forces.

Unlike the USSR, the western democracies HAD to rely on nuclear weapons because they couldn't impose the sacrifices necessary for adequate conventional forces in peacetime. Big armies require conscription and guns before butter. Naturally most people prefer that the national wealth go to them in the form of individual luxuries and amenities. And they'd prefer not to be in the army. Of course people in democratic states WERE willing to make sacrifices in wartime, when anybody could see the need. But often they didn't want to make sacrifices in peacetime. Since the western people could vote, they'd vote for those who did NOT make them serve or emphasize tank production instead of private cars.

Because the dictatorial USSR and its satellites COULD compel sacrifices, they were able to build huge armies. The USSR had more divisions in East Germany alone than the US had worldwide. In a purely conventional war, massive Warsaw pact forces would've rolled over those of NATO. The Soviets had a plan to overrun West Germany in 2-3 days and reach Paris by the fourth or fifth.

(True, NATO weapons had a slight qualitative advantage, but so did German weapons yet this failed to compensate for Soviet numerical superiority.)

And the US almost certainly couldn't have launched another D-Day later. Unlike the reich, the USSR had vast resources augmented by those of western Europe including all of Germany.

Wrong tense in your main question!!!

The only sensible answer is that, if the war had become nuclear, as would have been quite likely, then the whole world would have lost, and you would either not have existed (assuming you are of school age) or you and everybody around you would be living in abject poverty, and every day seeing people die of various nuclear-related diseases.

Assuming you mean in the nuclear age, everyone would've lost.

Depends... what would be the goals of that war, what are the victory conditions?

If we're talking about 20th century USSR and US trying to exterminate each other, then that would be a complete fail, unwinnable for anyone. Nuclear war? Everyone loses for sure if that happens... and in conventional war, if either side tried to take each other's territory, they would be easily beaten back, it would be a huge disadvantage for whoever is not on their home turf, especially given how far apart those countries are.

The whole world would lose, even those countries that don't get involved.

Because a nuclear war would destroy the environment and pollute it with radioactive substances.

NO ONE. There are no winners in a nuclear war, none.