"The end of the world" is a tricky term of quantify and there are a lot of variables (e.g., South America and Africa would likely not have come under nuclear attack in a US-USSR scenario — has the world "ended" if they survived?), but if you're asking, at what point did truly mutual assured destruction (in which the continuity of the nation would be severely questioned) come into existence, the answer is: the mid-1960s/early 1970s, more or less. It was only at that point that the USSR had the ability to project large numbers (e.g., hundreds to thousands) of warheads all the way to the continental United States with any reliability. One might wonder why you need so many weapons to do this, but keep in mind that the reliability of the weapons was not super high.

Note that if you lower the bar to, "at what point did the USSR get the ability to credibly threaten major US cities?", that drops the date to the early 1960s. Around the Cuban Missile Crisis, for example, the USSR had the ability to destroy a few US cities with some reliability — not total destruction, but still an unacceptably high amount of damage.

If you lower the bar to, "at what point did the USSR get the ability to credibly threaten the US' Western European allies?," that drops the date to the mid-1950s, when the relatively small Soviet bomber fleet came online (several hundred bombs were fielded by 1958 or so). These likely would not have been able to threaten the continental US with any real credibility, but they could destroy France, the UK, West Germany, etc.

An alternative approach is to ask, "at what point would a US attack on the USSR likely trigger a nuclear winter scenario?" Which is to say, if the nuclear winter hypothesis is true, in what year would the US war plans have been suicidal without them realizing it? There are enough unknowns here to be unclear, but by the late 1950s the US had the plans and means to unleash several thousands of megatons worth of weapons on the USSR in a short period of time, and if anything would cause the nuclear winter scenario, that ought to.

For information about the increasing size of the Soviet forces, see Podvig, ed., Russian Strategic Nuclear Forces.

It should be noted that from the mid-1950s onward, the US possessed the ability to totally destroy the Soviet Union. It had many more weapons than the Soviets did, and its extensive network of overseas bases with nuclear weapons gave it ready access to the country, and other allied bases (even non-nuclear ones) allowed it to field refueling planes that could increase the range of bombers launched from the continental US. By 1957 or so the US had plans in place to totally systematically the USSR in about 3-4 hours.