Actually, scratch this.
I haven’t got a clue what I did, but after trying another half dozen times or so it decided it would connect this time, and it is now working. Ran through the update and everything in a few minutes.
There was a minor scare, when the update operation seemed to wipe all my connected devices off the app, and any attempt to go to ‘Devices’ just asked me to set one up, but rebooting my phone appeared to fix that.
I have to say, this experience could have been a whole lot less hassle - I was close to sending it back.