Even most of the doctors don't speak English, there is an interpreter at the hospital that will very efficiently help. Also, all the forms to fill by the patience have an English version.
Even it is a private hospital you can use the Japanese health insurance.
I have visited the Musashino Tokushukai hospital. It was very good, but it's pretty far. For Musashino/Nishitokyo area.
https://www.musatoku.com/english/
Sanno Hospital is an English speaking hospital in Aoyama-Ittchome
Even most of the doctors don't speak English, there is an interpreter at the hospital that will very efficiently help. Also, all the forms to fill by the patience have an English version.
Even it is a private hospital you can use the Japanese health insurance.