While I am no historian, I am however familiar with the history of how we processed materials and made tools.

Let's just start with what the preferred method was, a drawknife well sharpened provides a wonderful finish, and some woodworkers today still prefer it to sandpaper. This is because by using a planer or drawknife to scrape the wood, the fibers are severed cleanly, rather than leaving a marred surface due to grit. The earliest example that I have is from Georgius Agricola, who briefly wrote about the use of this method in 1556CE in a book titled "De Re Metallica" a wonderfully detailed book on all the tools and processes used in mining at the time, it includes illustrations for pretty much every tool and workplace. He drew the drawknife being used with a shaving horse, and if I can find the page I'll include a the page number for my copy. Depending on the translator it might be a different page.

The British Museum has a drawknife from the mid 1st century. so it may be the case drawknives were used for finishing the wooden surface earlier than the 16th century. A search in wiki for a similar tool the plane/planer informed me it has been used since roman times for finishing wooden surfaces, and the oldest plane was found in Pompeii. And while the article suggests a plane may have been used in ancient Egypt because the wooden surfaces were smoothed with some manner of cutting or scraping tool, that's not really very good evidence as suitably sharp hand axes and knives can still be used for smoothing surfaces, we really can't know unless we find the tool.

As for sandpaper itself, the earliest form of sandpaper existed in the 13th century china, they used crushed shells or sand adhered to paper using a gum/glue of some form. I don't know the evidence for this one as I'm basing this off memory. Once again if I can remember what book I read that in I'll edit this, google isn't really helping me find a valid source for this one.

Sandpaper itself wasn't very common or widespread and other methods were used, I believe a few common ones were sharkskin (dog-fish skin), high silica containing fibers like rushes/scouring rushes and simply a piece of leather/cloth with an abrasive on it such as ground rottenstone or beachsand held on or lubricated with an oil or wax. That last one we still basically do with strops and polishing wheels, in fact brown polishing compound is a wax matrix holding Tripoli (rottenstone) particles inside it. u/idjet actually has a great reference for this being done in the 14th century! I found their post while trying to find a reference for rottenstone and the leather method.

Other methods of sanding/grinding existed, but not necessarily for wood as far as I know.