I wonder if you have something scanning incoming mail so it's delaying the process of download.

What Anti-Virus product are you using ? See if stopping it from scanning thunderbird files and scanning downloaded files/emails helps to improve.

Maybe there is an email in Inbox which is causing a log jam. Logon to webmail account via a browser and move the oldest not downloaded mail from the Inbox to another folder. This is the email you would expect to be downloaded first.

I'm assuming you keep old mail on server, if yes then try deleting a load of the very old mail. It can help in gmail accounts, so maybe worth a try. Sometimes servers can be awkward when people use pop - pop accounts often have a smaller quota on server because server expect users to delete stuff they already have downloaded. I have known pop accounts to start to function again if old stuff is removed.

I deleted the oldest not downloaded mail from the Inbox with Oullook, then the problem was solved! I deeply appreciate your advice.