Try this: open the “Security” app, then tap on “Permission” (I don’t really know if this is the translation in english, it is “Permessi” in italian :D, anyway, the red circle with a shield-shaped lock). In this section enable all the apps you want, then go back.
Tap on “Power Saving”, then tap on the icon with four little squares at the bottom, and enable the same apps here too.
So I had the same problem. None of my apps (Hangouts, Instagram, Inbox by Google) showed any notifications.
Above mentioned solution solved the problem - now the phone notifies me.
However, the LED notification light still doesn’t work for these apps. It lights on when I have missed calls. I have 220.127.116.11U Flyme.