The System Mail app makes it possible for Apple users to manage more than one mail account in one place, and it generally works well. However, sometimes iPhone or iPad Mail may not update automatically. You can access the Mail app, but there are no updated or retrieved emails.
Reset your iPhone
Have you tried turning it off and on again? Yes, it's a cliché, but it works a surprising number of times. Sometimes the mail app is out of sync with the network or email settings and the only way to restore the functionality of the app is to restart it. To restart your iPhone, go to Settings > General > Shutdown.
Once your device is up and running again, open the Mail app to see if it works as expected. If not, move on to the next solution.
Check your connectivity
Yes, it's a stupid question, but we have to ask. If the Mail app stopped working after the iOS 15 update, make sure you have a good signal, not in Airplane mode, etc.
Restart the mail app
Like other system apps, the Mail app might also crash or not work after the iOS 15 update, especially if it is left open or running in the background during the update installation process. In this case, you will need to force close the mail app to start it again. Swipe up from the bottom to the middle of the screen to reveal the App Switcher, then swipe up on the Mail app to close it.
Then launch it one more time through your home screen, Siri, or Spotlight. This sometimes helps get the mail app working again if it somehow freezes or stops working properly.