Well.... that went into pulling teeth territory and it is 3 hours of my life I won't get back, but I'm pretty sure I'm done now.
One phonecall to EE who, pretty much immediately washed their hands of me and said it was an Apple issue (here we go, I thought, pillar to post....) but did put me through to Apple where I was talking to a person within 30 seconds.
Three further online requests to Apple to call me back - and on each occasion my phone rang with a few (single figures) seconds of me making the request.
One thing I will say of both EE and Apple, it was all amicably and patiently sorted by four persons who were readily accessed and for whom English is their mother language, and they didn't try to get matey with me - BT, Talk Talk, et al, are you reading this?
I now have a new iPhone which works, and an old one which has been cleansed of the cormorant incident, ready for its next recipient.