If you experience a long delay in your incoming email, this issue might be caused by an over crowded IMAP server. All IMAP server have a limitation on how many connection is allowed to be connected to it at the same time. You can read more about it in here
If too many application (whether is web based or client based) tries to connect to it at the same time, one (or more) of them will be denied of access to the IMAP server, and therefore it can't get any new emails. As you can read in the link, Gmail has a limit of 15 connections. Mailbird by default will use 5 connections all the time, combine this with all the email client that you have in your mobile device, laptop, and web, then 15 connections will be jammed quite easily.
Now to mitigate the issue, you need to be able to control how many connections an email clients can have. To do this in Mailbird, open Mailbird menu, navigate to the account tab, then double click on one of your accounts. At the bottom you will see a connections slider, set this one to 2. Of course more connections is always better, but it can't be help, you need to limit them if you want to open a lot of apps at the same time.