This is a tricky question to answer, for the Egyptian The Book of the Dead does not have a single author. Rather, it was developed and compiled over time by many different scribes. Let's take a look at how that worked.
The Book of the Dead is a collection of spells and instructions designed to guide a soul through the afterlife and help that soul attain eternal life with the gods. This was not the first set of such funerary texts. The Pyramid Texts are the oldest, and they were originally reserved for royalty. Then came the Coffin Texts, which were inscribed on the coffins themselves and were still generally for the higher levels of society. In the New Kingdom, however, these kinds of afterlife instruction manuals trickled down to the lower classes, and The Book of the Dead was compiled from previous texts and from new innovations. Most likely, the book was put together by Egyptian priests and copied by scribes. It was not a static text, either, and there were variations between manuscripts.
One of the most famous examples of The Book of the Dead is the Papyrus of Ani, which was written by or for the scribe Ani around 1275–1250 BC. It is a beautifully illustrated edition that was discovered in Ani's tomb in the late 1800s and now resides in the British Museum. Yet we certainly cannot call Ani the author of The Book of the Dead, for its origins go back far beyond him, and we will probably never know for sure who first began this tradition of guidance into the afterlife.