iPhone 6 mail app delays
What's happening is that your iPhone is only fetching new data from the server only every so often, like every hour. To fix this, go into Settings->Mail, Contacts, Calendars->Fetch New Data and make sure that "Push" is enabled. Some mail services don't support Push, so the "Fetch" schedule below would be used. Scroll down and select "Every 15 Minutes" under the "Fetch" pane. That's a fast as you'll get notifications for mail that doesn't support Push, but you can always check for mail manually in the Mail app. Also, I should note that enabling Push and more frequent mail Fetching will result in worse battery life.