[GoNative]

If you're into the outdoors then I can't imagine why you would want to spend much time in Tokyo. I've been here 5 years and have only spent 3 days in Tokyo, I can't stand the place!
Since you're from the UK I would suggest Hokkaido as the summer weather here is much milder than down on Honshu which is incredibly hot and humid. I'm from Australia and I still find it almost unbearable down south in the summer months. For hiking and cycling in somewhat comfortable conditions I don't think you can go past Hokkaido.
Furano would be a great place to base yourself. It's a beautiful little town and you have easy access to some stunning mountains with plenty of hiking trails and it's an awesome region for cycling. Plenty of places up there you should be able to organise mid term rentals. Maybe even look at wwoofing it up north here.