Thanks for your answer. Actually the scenario you're referring to does not apply here. It is a personal phone with no content restrictions. I have identified a possible reason.
The reason why I believe the setting could not be changed is ... because did not allow the app to use wireless data at the app first launch.
During the first launch, I have noticed this panel :
We all know it, it acts as a "mini-firewall" for apps. At some point, I could have skipped this panel (maybe by switching apps, or receiving a phone call), then the system (or the app?) assumed that I did not allow the app to connect.
Usually, I could revert or change the setting from the Settings, but I could imagine that deleting the app or reseting all the iPhone settings could reset this choice, but the wireless data setting panel never appeared again.
The only way I found was simply to reset the Phone entirely, and download the app again.
To me, it appears to be a bug, yet I'm not certain it's for the app publisher or iOS.