I think the USSR was typically very keen to avoid a war for obvious reasons: they had a much smaller economy, exhausted population, significantly smaller military and massively smaller industrial base than the West.

America was more keen, but the logistics of actually invading the USSR would make any war essentially unwinnable.

Maybe if the nukes didn't exist, the USSR would have annexed western Berlin, but I don't really know how they would have managed more than that, even if they had wanted to.

I also don't think nuclear weapons would have prevented a war if the USSR was more aggressive or more close to the USA in military power.