The Mail App is Not Working on iOS 13
(ATTENTION: This issue has since been fixed)
Ever since iOS 13 released a few weeks back, several Apple users have complained that the Mail app doesn't seem to load/display new emails. This has been specifically observed in devices that have a global proxy or VPN set up to filter content.
Here's a post on the Apple Developer Forum that provides a detailed overview of the issue at hand. As this is an iOS-intrinsic issue, we will have to wait for Apple to fix it in its upcoming releases.
Meanwhile users can try 3rd-party mail apps like Spark, Outlook & Airmail to keep in touch with emails.