I read this story as part of an anthology sometime in the early '90s. The protagonist survives a nuclear war but when he comes out of the bomb shelter he is "beamed" up to a mysterious organic starship, along with other survivors from the Earth. I don't really remember any other details other than the protagonist's wife committed suicide as soon as the came out of the shelter.
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).