the third Reich was a very unstable regime. If they would have won the war, they would either become democratic - which is no difference from now-, or become like the USSR or NK.
For a long time it appeared to people that the USSR was on top off things (sputnik, first mane in space etc). Eisenhower knew that and lost elections over it. We now know the technology gap with the US was ever growing in favor of the us. Its always the same with these totalitarian regimes. The box looks good as long as you leave it closed.
Anywayz,
A lifeboat in space is much, much more costly as a lifeboat underground. A lifeboat underground is more difficult to destroy if build well. Yes ICBM's can reach the moon.
The effects of a nuclear winter far exceeds the 1-3 years of dust. Global climate and environment will be destabilized for decades or maybe centuries, dependent on what weapons used.
there are so many variables noone can predict what would happen.
- It all depends on what kind of nuclear weapon used and how many
- radiation depends on that too
- the extinction of many lifeforms is unpredictable
- the forming of large desserts
- the fierce storms
- the possible lack of an ozone layer
Its possible you have to maintain humanity underground and feed it for half a century before you can even start to rebuild a significant agriculture above. That is, if your shelter is not destroyed.