I had the same problem and finally figured out the solution!
In short, you need to:
- Go to Settings -> General -> Language & Region -> Preferred Languages -> Add Language...
- Add English (UK) or English (US) as a preferred language
- Ensure that English (UK/US) is ranked higher than Japanese in your list of preferred languages
Why This is Happening
Apple's translation software only supports UK or US English. If you are using any other English variant, translation is not supported.
In my case, my preferred languages were:
- English (Canada)
- French (Canada)
- Japanese
And since neither English (Canada) nor French (Canada) are supported translation languages, Apple Music was showing my 3rd language preference (Japanese). What makes this more confusing is that my Settings app simply stated "English" as my system language, not English (Canada).
I've now changed it as follows:
- English (Canada)
- English (UK)
- French (Canada)
- Japanese
And Apple Music now correctly shows lyrics translations in English (while still maintaining Canadian English as my main system language).