Right, I have made some progress but got stuck again.
I have never had two step verification or two factor authentication enabled.
On my iPhone and iPad under my iCloud Password & Security settings there was no option to enable two factor or two step. On my iMac and and MacBook in Settings once again the option to enable was totally missing. Logging into the manage my Apple ID website through a web browser there was no option to enable either.
Some time ago I tried to set up two step verification. Because I had recently changed my password I had to wait three days before I made major account changes, including two step. Three days later Apple sent me an email to tell me it was ready but I never followed through. This seems to have locked up my account.
I clicked on the web link in the (now very old) email and the option to enable magically reappeared on the manage my Apple ID website. I enabled and then disabled two step verification.
I now have the option to enable two factor authentication on my iPhone and iPad but for some inexplicable reason I get the error message 'Some of your devices are not ready for two-factor authentication. If you continue, you will have to add a six-digit verification code to the end of your password any time you enter your password on an old device'. This is despite the fact all my devices are running iOS v9.3 and OS X 10.11.4.
Apple. It just works.