The problem sounds like the standard IMAP problem:
By default IMAP is designed to look only in your INBOX for new mail. Especially when your server supports IMAP push/idle command you get an instant notification for new mail in your INBOX.
All the other folders are not checked by default. In Thunderbird you can selectively add folders Thunderbird should check for new mail (see folder property dialog) or you can specify to check for all folders by setting
mail.check_all_imap_folders_for_new = true
in the Config Editor. Anyway the additional folder will only be checked every x minutes, depending on the IMAP account settings you have specified.