Deep sleep issue

I have an Iphone 5. Lately, it has stopped giving me notifications for any apps that need internet to work while the phone screen is locked. This goes for Imessages, Viber, any email notifications. Once I unlock the screen, the notifications flood in. DND is off, text messages and calls come in fine even with a screen locked.

I decided to troubleshoot the problem and started sending Imessages from another number. When the screen is unlocked the notifications are instant. However the moment the screen is black, there are no notifications, nor I see the delivered sign for the imessage. Same goes for VIber messages. When I unlock the screen, notifications flood in.

Tried restoring network settings - didn't help

Tried setting up as new phone - didn't help

Used a different sim card - the problem still occurs.

Does anyone have an idea what is causing the phone to go to deep sleep and disconnect from the internet when the screen is off?