There is already a word for converting something to English:
Anglicization /Anglicisation (AmE/BrE)
1 : to make English in quality or characteristics
2 : to adapt (a foreign word, name, or phrase) to English usage: as a : to alter to a characteristic English form, sound, or spelling b : to convert (a name) to its English equivalent <anglicize Juan as John>
It is almost only ever used to refer to the conversion of words or names, but it only needs a slight expansion of its meaning to cover the conversion of all business communication in a company.
A slight broadening of an existing word, in a way that is consistent with its history and existing usage, seems much preferable than coining a new, awkward phrase.
However! I'm not a Japanese speaker. What is acceptable depends on who has to accept it, so what word is acceptable for this idea is ultimately up to the people who use it. A Japanese speaker unfamiliar with the "English"/"Angle" transformation would probably find "Anglicization" unintuitive, and the n following the sh in "Englishnization" may well be roll off the tongue better for someone coming from a language in which a syllable-final sh doesn't exist and the only syllable-final consonant is n. As it is Japanese workers and companies who are the target demographic that is being sold on the concept, what a Western English speaker thinks might be irrelevant. Not knowing the agenda of Rakuten intimately, I can't be certain.
So, the short version:
- If you are asking what is most acceptable to a Western ear, either "Anglicization" or "Englishization" is what you want.
- If you are asking what is most acceptable to a Japanese ear, I cannot say for certain – but I would not be surprised if "Englishnization" becomes most accepted.
- If you are asking what is most acceptable to both, then you want to compromise and choose "Englishization", no n.