Did you already check if "Cellular Data" is activated in Settings/Cellular?
Also explained in this article:
When cellular data is off, all data services use only Wi-Fi, including email, web browsing, push notifications, and other services. If cellular data is on, carrier charges may apply. For example, using certain features and services that transfer data, such as Siri and Messages, could result in charges to your data plan.
Message was edited by: Ingo2711