I think I have solved the issue. I deleted the last shown photo from within Photos on my Mac (the one at the bottom of the window). None of the files in iCloud would download after the date of that last photo, but after removing it, closing Photos, and opening it again a few minutes later, they all appeared.
Just to test, I took a new photo with my iPhone and closed Photos on the Mac. I waited a minute and the photo appeared within Photos just like it should.
Thanks again for all your help!
