Usually, it’s something simple: either your internet is weak, Background App Refresh is off, or Mail isn’t set to fetch new data. I’ve had this happen when “Fetch New Data” was set to manual, so emails didn’t update until I opened the app. Also, check if Low Power Mode is on, your account password is still correct, and Apple’s Mail servers aren’t down. A quick fix that works for me most times is just toggling Mail off and on in settings or restarting the phone.