Edward,
With all that you report, my first thought is that it is on their end of things. It is a rare provider, such as this, that will be entirely forthcoming.
One thing to try, before moving on the suggestion below, is to delete the file named MessageUidsAlreadyDownloaded and the MessageUidsAlreadyDownloaded2 files from this account's folder. This POP account folder will be found, with the Finder, at Home/Library/Mail/your POP account folder. Do this AFTER quitting Mail, and then relaunch Mail, and test. If these files were somehow corrupt, they might cause this timing out.
If no help, then one test to conduct, to help eliminate some things, would be to create a new User account on your Mac, and set up your accounts in Mail, in that User account. See if there is any difference.