My iPhone 5S updated fine via WiFi to the GM of 10.0.1 (public beta).
However, before doing the iOS 10 update, I recommend:
1) Update iTunes to the latest version.
2) Backup your iPhone/iPad locally (+/- iCloud backup).
3) Archive the backup, so it can't get erased. (Preferences --> Devices --> right click on most recent backup to archive
4) Download all your photos somewhere safe
5) Update
Apple doesn't mention doing #4, but I think this is necessary, because if your phone really gets totally bricked or needs a wipe, you've lost your photos as well.