Team -- I have a new iPhone 6+, and I am having the "PUSH" issue too. I have 4 phones of various flavors and OS levels. All the other devices are alerting as they should; only the 6+ is not alerting until I actually enter the app. This is a corporate Exchange account (2003).
As Murphy would have it, this is my VP's phone and he has VIP alerts setup from our CIO\CEO and system outage alerts. So needless to say, PUSH is critical.
Update: I have another user on a new iPhone 6 with the same issue... exchange mail not pushing...
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On a side note. Not sure if anyone mentioned this earlier, but Gmail no long uses PUSH on free accounts. It will apparently continue to work on your old device until you have to reload the app. If you get a PAID account, gmail will offer PUSH.