Do you think it is a notification-related problem, or receiving problem?
The subject of your posting speaks about delivery problem, but you describe only notification problems.
Do you have a situation when the message(s) had been received on the phone, but no notifications were shown?
In the mean time, you might want to check the settings related to synchronization and notifications:
Settings -> Mail, receiving:
1. General -> System Settings -> Sync even when "Auto-Sync" is off
[Or make sure your "auto-sync" in the Android settings is on].
2. Scheduled Sync -> Scheduled Sync is ON
[I might be wrong, but I thing even with Push enabled, this is fine to have.
And you said you checked it.]
3. Scheduled Sync -> WiFi in sleep mode is ON
4. (optional) Check Mail on Events (WiFi connection)
Make sure that you didn't accidentally unchecked any of these in the Per-account settings (long push on the account in the account-list view mode, -> Account settings and folders)
5. Under "main" Settings -> Message notifications,
You might want to turn on "When the app is active".
I am thinking that if Aquamail is the only app that you are using, and you leave it in Aquamail, - it might consider the status of the App as "active", - even though intuitively I'd think it shouldn't be.
6. Also make sure that Push is enabled in the account settings, and for the Inbox.
(See FAQ item "How do I enable IMAP Push ?")
http://www.aqua-mail.com/?page_id=2277. While reading FAQ, the next item "Notification icon in IMAP Push mode" might also be useful to you.