iPhone delivers SMSs but says they weren't delivered

I get a "message send failure" every time I send an SMS, although actually they deliver okay as the recipients receive them.

When I send the text, a green loading bar appears on top and it gets stuck 10% before the end. After 2-3 mins I get "not delivered" red text next to it.

But actually the recipients receive the text as soon as I send it. It's an SMS, not iMessage.

So the iPhone thinks they weren't sent - and gives me the option to retry - whilst actually all is good.

This behaviour only started happening today and I can't think of any changes I would have done that would have caused it.

The same happens when I try to send a message using "Messages" app on my Mac.

Restarting of app and phone didn't help.

Switching iMessage on/off didn't help.

Any ideas?