How can I get notifications to show up on my Apple Watch again instead of my iPhone?
I restarted my iPhone to try to fix this. Then I took a look at the Messages settings for my iPhone in the Settings app. It turned out that Messages was logged in to the “old version” of my Apple ID—a couple of months ago I changed the email address of my Apple ID and Messages was still showing the old one. I logged out and back in, using the new email address this time, and now I’m getting iMessage notifications on my watch again.

Since these notifications were working initially, though, I’m guessing that the Apple ID mismatch was a red herring and that restarting was what fixed the problem.