This is highly tentative, but Megan Lindholm, also known as Robin Hobb wrote several stories featuring sentient ships. In the novel Alien Earth (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 trilogy). So maybe the story you remember is one of her short stories (which I haven't read).
user56