The most straightforward way for you would be to obtain a bachelorfs degree in the UK first.
Please note that a bachelorfs degree in Japan usually takes 4 years, and if you are thinking of enrolling in a Japanese university, you would need to study the language (normally 1.5 – 2 years in a language school) first to be accepted. On the other hand, you could study in an English-only course as well. But whether counselling major is available at a such university is another question.
You might want to consider your long-term career as well; if your goal is to take up counselling-related job in Japan, fine, learning the language and majoring it here might make sense. However, if you are thinking of returning to the UK and working back home, then a degree here might not be worth it. (Personally I would prefer to study a topic like counseling in the UK, rather in Japan. Occupational doctors and counsellor work is limited here, I feel.)
I do not know what kind of teaching career your partner is thinking about – English teaching? Does your partner have an idea about how much that would pay and the long-term prospect of it?
Also concerning visa, you need to think about timing: your partner finds work, the employer-to-be in Japan starts the visa process, then for gdependenth (unless the employer is extremely cooperative) it would be up to you to start the visa application process only after your partner has theirs granted. So there might be a time lag/waiting time.
If you go the gstudenth visa path, you need to apply with the school, get accepted by the school/university, pay in the tuition (for one year or so I believe at least), then the school will start the student visa application process for you. Youfd need to start the admission process pretty early.
For gstudenth visa application, there is a requirement for you to be able to show that you have enough funds to support your studies and living while in Japan.
For information on any aid to student, look at JASSOfs website:
https://www.jasso.go.jp/en/index.html