As far as the Koran is concerned, Muslim legend claims that it wasn't written down at all for decades. It was just passed orally from one student to another, memorizing pieces. Given the nature of oral traditions around the world, it's not as crazy a legend as it sounds to modern people. Would Mohammed have had to read for himself the stories of Christianity and Judaism? He was the husband of a wealthy merchant, so he undoubtedly could have had the stories read to him.

In Joseph Smith's case, he is supposed to have dictated the Book of Mormon to a friend based on what he read on some gold plates. It's hardly a work of great literature, charitably it could be called ponderous. Of course, this is the same fellow that "translated" an Egyptian text by simply making up a story about Joseph. Serious scholars treat all his claims with extreme scepticism.

Were they actually illiterate? I'm unaware of any source that confirms or refutes it for either. But it's certainly not impossible.