It's an iOS 13 (and 13.1 to a lesser extent) issue. There's currently no fix for the stock Mail app. Some people have tried to uninstall the stock app and when they reinstalled it, it was even worse.
I find that my alerts do come in ok apart from the first lot when I turn my phone on in the morning. They either don't come in at all, or come in silently. Other notifications also come in silently for a while though, so I think on top of all the Mail issues there are also some more generic Notification issues.
13 and 13.1 seem rushed.