The first step was to find some pictures on the internet and pull together measurements. Lots of websites have useful information, including these: for photos of simple windsor chairs, for dimensions, or for detailed drawings and tips on construction.
After testing a few different woods, I settled on boxwood–it has a fine grain, is flexible and strong enough to withstand turning to small diameters, and the color is neutral but not bland (in case the chair is not stained).
The final pieces:
The most challenging part of the chair were the various spindles–for the legs, the stiles on the back, the spindles connecting the legs… you get the idea.
The spindles were formed on a lathe using a combination of roughing down to about 2 mm, using a graver as a scraper to thin it another half millimeter, and then sanding with 180 grit sandpaper and fine files to get the spindle down to a mm or less, depending on the piece.
The drawplate was used to get the spindles down to their final diameters.