I had probs with MMS on Three in Ireland.
In the end a full reset with their sim inserted worked. I then learned not to do anything with the network settings, i.e. don't run Network Setup or tinker with the APN settings in the Diagnostic tool. I also have not restored any backups, just started afresh.
This problem is now solved for me.
The latest firmware (12070) and the latest Network Setup app (22.214.171.124) creates the correct APN settings (internally) for MMS and Data.