For the benefit of anyone else doing this in Japan, these are the documents I had to submit. It will be different depending on circumstance, in particular I think the author may also be born outside the UK which may require other/more documents.
Anyway I had to submit:
Certified copy of my birth certificate, order from the UK general register office.
Original (not photocopy) of Japanese family register and translation.
Certificate of acceptance of notification of birth, original and translation.
Original Marriage certificate and translation.
Colour photocopy of every page of my childs Japanese passport.
Copy photocopy of the passport of an American or British citizen who confirms that child is mine.
The process is pretty unclear, and in general you seem to have to just keep submitting documents until they are satisfied.
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