My 4th generation Apple TV was not showing a new shared album I created. I ended up fixing it by going into
Settings ➔ Accounts ➔ iCloud
on the Apple TV and signing out and signing back in. Then turning photo sharing back on (it’s off by default when you sign back in):
Settings ➔ Accounts ➔ iCloud ➔ iCloud Photo Sharing ➔ On
Very lame I know, but it worked!