Do you get notifications for new email from your other Gmail accounts?
I just tested with two Gmail accounts I use. On my computer, I sent a message from account #1 to account #2, waited until I heard the tone on my iPhone and saw the notification on the screen, then sent an email from account #2 to account #1. Both times I heard the tone and got a lock-screen popup.
In Settings &gt; Gmail I have this configuration: