No nuclear weapons when would the us and user go to war

Let's say for what ever reason nuclear weapons where never invented, when would ww3 start between the us and ussr. As far as I can tell both the us and ussr when pretty exsasted (asecaly after the us is forced to invade japan) so I don't think it would right after ww2, so Maby 20-30 years after, the Berlin crisis , the 80's?
 
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