Robert Silverberg

From his Nebula nomination for his short story The Pope of the Chimps [1]:

Nominated for Nebula Award for best short story, 1982 (sic). This is the moving story of a long-term chimpanzee sign-language project. When one of the scientists studying the chimps gets leukemia and talks to them about death, the chimps develop a religion. Humans are above chimps, and God is above humans. When chimps die, they become human; when humans die, they go to God. The time period is unspecified (apparently late twentieth century), and there is no technology mentioned. In fact, some might say it's not science fiction at all. An interesting side note is that Silverberg, himself atheist, assumes a more or less Christian perspective for the story. Another odd thing is that the first person narrator is never given a name.