I'm having exactly the same issue, and I just finished the PS4 version an hour ago. I have good endings for all four routes - all show checkmarks - and I just can't get New Game Plus to show up. Nothing has changed on the Extras menu, and "New Game" just starts a regular new game.
The only thing I'm missing are a few CGs, but from everything I've read you don't need to unlock them all for NG+.
Edit: I can confirm that doing Julian's route last was the issue, and that replaying another route unlocked NG+. Thanks for reaching out on Reddit for the help!