I had the same problem with no incoming mail in Apple Mail, but it only affected my Outlook.com incoming mail. Others worked fine. In the Apple Support Communities, I found that the workaround was to disable the avast! Mail Shield, but turning it off defeats the purpose of a virus protection. A later post to that same thread said that disabling the Mail Shield, closing Mail, restarting Mail, then re-enabling the Mail Shield restores full protection and allows incoming mail again. I haven't tried that yet, but others might try it.
Meanwhile, Avast needs to figure out what they messed up in the last update that causes this problem.