Mine collects new emails from gmail by itself. Set to push and 15 min intervals

Yup, collecting at intervals is Fetch (aka Poll), not Push.
That is what it falls back on when it can't do Push.

To see if you are getting real Push then go into

Settings ➡ Mail, Contacts, Calendars ➡ Fetch New Data ➡ Advanced

and check what it says next to your account names.
It will say Push, Fetch or Manual.

Yep, set to "fetch"

Yep, set to "fetch"

oops, double post. Rubbish these iPhones 🙂