For those of you (like me) who were still having this issue on Big Sur (email slow to arrive or not arriving at all in Apple Mail for Microsoft Exchange 365 account), this may help, particularly if you have a large number of emails on the Exchange server.
After deleting and reinstalling the exchange account, as advised in this thread, the problem got worse for me. I had to log out of my account on my laptop (16” MB Pro) or restart it to get any exchange emails at all, and then up to about 10 new emails would come in at once and then no more until I logged out/restarted again. Quitting the mail app and reopening it didn’t help, only complete log out/restart. Note that my iPhone and iPad were getting the new emails without any issues
I opened the Apple Mail “Activity” box that can be used to show what the mail app is doing (drop down menu in toolbar under “Window”). This revealed that there were still over 36,000 emails to download. Obviously a hang-up from uninstalling and reinstalling the email account. Clearly it was taking a very very long time to synchronise. Note this was 3 days after I had reinstalled the Exchange account and almost the entire time my laptop had been switched on and connected to internet. I noticed that virtually any mail related activity seemed to really slow the sync process.
I was able to leave my laptop switched on and active over the weekend on a fast internet connection, and I disabled the sleep process so it could download the emails more quickly. Once the number of emails to download fell below about 30,000 the new emails started coming in, although still a bit delayed. As the number continued to drop the emails came in quicker and quicker until there was no delay below about 8,000 remaining.
The mail app has now completed the synchronising process and I am receiving all emails without any delay. Therefore I assume that reinstalling the account worked as hoped in solving my original problem, it just took a few days to sync everything back up.
