Hello Mummycorn,
We see you're having issues with the Caller ID of incoming calls, and we'd like to help out.
As it's happening with every call, this may be related to the mobile carrier. It sounds like they attempted something already, but you may wish to try following up with them for further investigation to see if they can re-provision the account.
You've tried some good steps already. Be sure to also check on things like the date and time (Settings > General > Date & Time), to ensure that's accurate.
Some users have found success by temporarily switching the country of the device in Settings General > Language & Region, and then try switching it back to your normal one.
Also, try removing the SIM card from the device, and then reinsert it: Remove or switch the SIM card in your iPhone or iPad
Take care.