ios new mail doesn't show until the mail app reopens

• Is this your corporate iPhone? Then please contact the system administrator of your corporate
• If it is your personal device, in that case do you have any profile installed by your corporate/organisation/enterprise/School Management? Then also, please contact the system administrator of your corporate/organisation/enterprise/School Management.
• Else, delete and add the mail account in the IOS mail app. It is a matter of 1-2 minutes only.

You are aware that there are two settings (Fetch and Push Mails). What you have set is Push as per your post. I am just sharing my knowledge about the entire process, see if it matches with yours. Watch the GIF below

Google, AOL, Yahoo - Only Fetch mail Option is available

iCloud, Exchange, Outlook - Both Fetch and Push Mail option is available

1. Push Mail: Whenever possible the email client will push emails to all your devices.
2. Fetch Mail: You need to set How you want it to be fetched, Automatically, Manually, Hourly, Every 30 mins, every 15 mins
   1. Automatically: will fetch when the iPhone is powered and on WiFi
   2. Manually: with fetch when you open email, open Inbox and swipe down
   3. You get an email notification as soon as the client-server receives an email and your device is connected to the internet, irrespective of whether the mail is in your device's Inbox or not