Looking for an iOS app to announce the time
As Joe indicated, the reason why you have not found an app to do this is because apps simply can't.

Backgrounding apps can do very little, and even things that they can do are limited to generating a notification which you must acknowledge in order for the app to actually carry out it's task. For example; Skype can background and detect a call, but you must still acknowledge the notification to open the app, and have anything meaningful actually occur. Additionally, backgrounded radio stations / audio streams can continue to stream, and even change channels or support some feature that responds to the "skip forward" and "skip backward" media controls, but if the network connection is broken the app is suspended and will not do anything further until you re-open it.

Sadly, this is also not part of the accessibility features built into iOS, perhaps you could file a bug and request the feature be added in a future software version?