For iOS, Secure Email alerts and requests do not appear until the Secure Email app is brought to the foregound (for example, new email message alerts and PIN change requests). As such, users aren't immediately aware of the alerts/requests.
Please explain why other apps are able to send push notifications to my iPhone lock screen. What is described as a limitation of iOS does not seem to prevent many other apps from sending push notifications even if they are not running in the foreground?
Secure Email uses your Exchange server to send emails, but Exchange does not know how to send APNS notifications. You can read all about it here: http://developer.apple.com/library/ios/#documentation/NetworkingInternet/Conceptual/RemoteNotificationsPG/ApplePushService/ApplePushService.html#//apple_ref/doc/uid/TP40008194-CH100-SW9
Secure Email uses a third party vendor. Here is more information about how iOS features and suport is different than support for Android. http://blog.nitrodesk.com/2012/10/02/touchdown-for-ios/#comments
The very first point deals with the push notifications, including Emails.:
1. Push: Note, however, that Apple does not allow applications to push data in the background, hence updates will not be pushed to the device if the application is not active. Once started, the app will start bringing in the updates and will continue to do so as long as the app is in the foreground and active. This is a change from how things function in Android.
Since applications cannot periodically wake up on iOS, there is no concept of periodic polling either on iOS. Peak Times is another concept which will go away, since there is no background syncing.
Symantec App Center, Secure Email on iOS