So it turns out Apple’s “Theft & Loss” cover isn't actually handled by Apple, it's handled by AIG, whose site is incredibly buggy, and among other issues won't accept my payment (tried multiple cards & browsers).

Apple support are unable to help with this and can only suggest contacting AIG directly by phone.

So when you get your phone stolen, the only way to arrange a replacement via Apple's “Theft & Loss" cover is to… make a phonecall.

It's been a fun evening.