On the third time I turned off my router it worked. What seems to screw things up is that I shut down my phone each evening as I do other devices. In the morning, it goes into panic mode and while saying it's connected to the internet, it isn't. 4G does connect to the internet, but then I get asked to log into my apple account, which it fails to do over 4G, seemingly requiring the same connection (wifi) as it had previously. Argh! Really starting to think about returning to Android. I'm sure some of this is to do with iCloud (something I don't want or need). After I connected and signed in just now, it reverted some of my iCloud settings, ie turning photos back on! Apple need to make this work without iCloud and make it far easier to trouble shoot the issue, such as providing decent error log messages for a start (which it's possible to find and read!)
Ok, rant over! Thanks ProustGiulio for the reminder about turning the router off again. It was next on my list but always worth mentioning. I guess it was trying to use the same local IP address and/or channel, which got it confused. It's a pain you can't reset this from the phone if that was the problem. Note that I had already reset my network settings earlier.