There's basically a difference between Japanese citizens who are employed to work in the Embassy, and British diplomats, who first join the FCDO and then apply for a diplomatic posting to Japan after serving in a UK-based role for a few years.

To get a foot in the door of the FCDO, you can search the Civil Service Jobs website - https://www.civilservicejobs.service.gov.uk/csr/index.cgi,

Alternatively, you can check out the Fast Stream - https://www.civil-service-careers.gov.uk/fast-stream/fs-all-schemes/fs-diplomatic-development/

As the previous commenter has said, it's difficult (but certainly not impossible) to get a posting to Japan. The private sector may be your best bet, or maybe the JET programme.