Some spells are in Greek, like Episkey or in Aramaic like Avada Kedavra or in another language like Alohomora.

Spell names don't really mean much. They are a designation more than anything, but they do help with control when you learn, similarly to hand movements. They are not necessary to perform magic though, so translating words a different way would not affect what the spell did, outside of the Floo network, where pronunciation determines where you go.

So them they could use English to do the spells too? (I understand that eventually you don’t have to say it because it’s in you and you don’t need the words to control it more).

With proper hand movements, control, and concentration, I don't see why a foreign student, Japanese for example, couldn't use English words to cast magic.

However, I don't think saying the spells in your native language would be effective. Control and proper pronunciation are still important to learn magic, so Ilvernorny would not teach magic in English. Uagadou or Mahoutokoro, on the other hand, might have success teaching magic in English, as long as the control, pronunciation, and hand movements are preserved to ensure proper learning. One of the schools, I believe Mahoutokoro, even teaches magic without a wand. All these things are just catalysts that with proper control can be omitted or changed.

EDIT: Correction, it is Uagadou that teaches wandless magic.