I've got a Sturmey Archer cog that's noticeably but not severely worn. Is it OK to just turn it around and keep riding it, or does it need to be replaced? I guess 15 years of service is pretty good for a cog, but still, waste not want not.

Should be OK. If it is a dished cog, your chainline will shift about 5mm. A flat cog will change nothing by flipping.
On the other hand, cogs are cheap.

^^ this,,, 13,14,15t, (in my past AW3 findings) are flat, 16t+ are dished.
have 2 spacers on the hub? moving those can keep chainline close,
after flipping cog..

Although now Sun-Race S-A may have expanded the range of flat cogs
above the 3 mentioned.

..

It's a dished cog. Good point about the chainline.