Documentary evidence for the use of wood screws dates back to at least 1556, when Georgius Agricola, the author of De Re Metallica, notes that in the construction of large bellows:

Some people do not fix the hide to the bellows-boards and bows by iron nails, but by iron screws, screwed at the same time through strips laid over the hide. This method of fastening the hide is less used than the other, although there is no doubt that it surpasses it in excellence.1

To be sure, the use of screws would have been unusual. The threads of each screw had to be cut by hand, a tedious and time-consuming process. But Agricola's screw is startlingly similar to what you will find in a modern hardware store. While early surviving examples of hand-cut screws have square ends, the screw illustrated in De Re Metallica has a sharp end and round head, very much like a modern one.

Notes

(1) Aricola, Georgius. De Re Metallica. (Hoover, Herbert Clark and Hoover, Lou Henry, translators.) Dover Publications, Inc., New York, 1950. p. 364.