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Are you sure there is no issues with your data plan or data connection on carrier side ?
If it is not working, then lets try resetting network settings. Go to Settings> General > Reset and tap Reset Network Settings. This will reset all old network settings. You will also be disconnected from WiFi. After resetting network settings, go to Settings ? Cellular Data and enable it. Make sure Data Roaming is off in there as well. Does it work now ?
If that didn't work, then connect your iPhone to your WiFi network and lets configure the APN (accesspoint). On your iPhone, go to this website on Safari --> http://www.unlockit.co.nz
Follow the onscreen instructions on that website to add the APN of your carrier. Then check if it is working or not.
(Note : You can also call your carriet network to get the APN details and add it to your iPhone in Settings > Cellular Data)
Please let me know if that didn't work, so we can continue troubleshooting