Also having this problem. Prior to iOS 11, phone received push mail in Gmail app and fetched mail in Mail app not long after. I have been running a Gmail account through iOS Mail as well as Gmail app with same account. I run the Gmail app for push benefit because of my job. It always worked. Always same results on WIFI or cellular data, locked phone or open. After the update, Gmail app no longer retrieving mail unless I am IN the app. iOS Mail app fetching mail without a pattern. One will show up immediately if in the app, within ten seconds if app is in background, but five emails will take an hour if the app is closed and the phone is locked. No locked screen notices AT ALL despite that setting checked in notifications. I wake up in the morning to no email alerts at all, but then I enter the app and find seven waiting for me from hours before. The phone is not behaving according to the settings.
As for the phone overriding the ability of the Gmail app to even function at all, I don’t know if that is a Gmail software bug or an iOS 11 ****** move. But this is deeply aggravating and hurting my actual livelihood. I tried all kinds of settings changes and toggling. Powering off. Restarting. Removing accounts and apps and reloading them. Nothing. Went for the ultimate Hail Mary and factory reset. Guess what? iOS 11 is still on the phone despite the factory reset. So I lost everything on my phone except the terrible new buggy software. Beware.
<Edited by Host>