RetireJapan

I'll check out ap-advisors as they do list it as a service on their site. They don't have a price though, which is usually worrying.
I'm not sure it's worth me paying more than a few thousand yen for.

The best interest rate I'm getting on any of my uk savings is 1.55%, so even the 3.65% that NS&I's instant access is offering looks appealing.

I'd leave your parents' address for now. And then, you can always use a virtual UK SIM card for confirmation, if you don't have a number.
Use the numbers from https://anonymsms.com/. They have UK numbers, and it's a free service.