This is highly tentative, but [Megan Lindholm, also known as Robin Hobb](http://en.wikipedia.org/wiki/Megan_Lindholm) wrote several [stories](http://www.isfdb.org/cgi-bin/ea.cgi?Megan_Lindholm) featuring sentient ships. In the novel [*Alien Earth*](http://www.isfdb.org/cgi-bin/title.cgi?2442) (1992), a sentient spaceship is one of the main characters, and another main character is an earthman who was taken off-earth (along with others) when the Earth was slowly succombing to ecological disaster. This is probably not the book you remember since being taken off-earth is part of the backstory, not the main story, and there was no nuclear war, but the author used the sentient ship theme in other stories (the [*Liveship Traders*](http://en.wikipedia.org/wiki/Liveship_Traders_Trilogy) trilogy). So maybe the story you remember is one of her [short stories](http://www.isfdb.org/cgi-bin/ea.cgi?Megan_Lindholm) (which I haven't read).