Just noticed this thread, so posting here too:
I have had issues on my XS on 13, now on 13.1.3 (ever since 12.4.1) regarding e-mails (two Exchange accounts, Office 365) not downloading in the background, both using the stock mail app and Outlook. I am now using Outlook on my XS and Macbook and Mac mini, both on Catalina.
In Outlook (the stock mail app might have similar settings) on both the Macbook and Mac mini, I went to Outlook - Preferences - Email - Reading, then under 'Mark emails as read', ticked 'Only when opened in separate window', kept the two other alternatives unchecked. Restartet Outlook on the desktops and on the XS.
For the last 1-2 days e-mails coming in on my XS are notified perfectly fine, downloaded in the background, i.e. no delay/issues in e-mail flow. Also, all is fine on both the Macbook and Mac mini. Hoping it will last....
In case you are running Outlook on a PC, you might try the same there. If no other Outlook connecting to your mail accounts, I have no suggestion.