Highly doubtful that you could obtain a work visa for Japan using your specialty unless you can speak Japanese fluently. You may be able to obtain a work visa for something like English teaching, which many people do, and potentially use that experience to make connections that will enable employment outside of English teaching. One long shot for you would be to try to get employed at a Japanese corporation in the UK that is involved in areas of public health (pharmaceuticals, etc.) and see if they will entertain the idea of transferring you to Japan on an intra-company transferee visa.
English teaching is sort of the low barrier-to-entry way of getting into Japan, but it can be a trap.
Public health is not my specialty, but if your qualifications make you attractive to a Japanese pharmaceutical company, that could be one possible route to living in Japan.
As a long, long shot (but maybe not that long... I don't know the industry that well) you might try contacting LinkedIn recruiters in Japan to see if they can help place you in a Japanese company. I have heard that Japan is experiencing a labor shortage, so maybe your conversational Japanese is enough to make you sufficiently attractive for a Japanese healthcare company to go to the effort of getting you a visa and hiring you.