Apple - How to enable push notifications of emails?

You're talking about the notification settings for Mail, not whether push is on. To see new mail messages in the Lock Screen:

Settings > Notifications > Mail > Make sure "View in Lock Screen" is turned ON.

If you're concerned about your mail's push settings, those are in:

Settings > Mail, Contacts, Calendar > Fetch New Data > Make sure "Push" is set how you want it.