I understand how to set the cover ("key") photo of an Apple Photos album, but I can't see how to change the cover photo of a "shared" (copied) album.
How do I set the cover photo of a "shared" Photos album?
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With shared iCloud photo albums the key photo will always be the photo that was uploaded last.
Update: This is still valid for iOS 12.1.2.
Actually, you can. Open an image to full view in the Shared Album, then in the menu select Image -> Make Key Photo. The key is to open the image to full view, it's not possible to select a Key Photo from the thumbnail view.
There's no way to do this. Though you can temporarily set a "key photo" (though a surprisingly awkward procedure), that will reset to the first photo in the album as soon as any photos are added.