Why does the Old Testament (in the English versions) use Greek names instead of Hebrew names?

Well, this won’t sound pleasant, but:

The normative version(s) of English, as with most European languages, are culturally influenced by Christianity more than other religions. (Jewish Englishes will in fact use Hebrew names, just as Yiddish does. But standard English unsurprisingly uses Christian forms.)

Christian knowledge of the Hebrew Scriptures was mediated through the Septuagint, the Greek translation of the scripturs—either directly (Orthodox Christianity), or via the Vulgate  (Catholic Christianity). Because of this, English (and most European languages) use Greek versions of the names: Moses (actually Mōysēs in Greek), Jerusalem (Greek Ierousalēm), Solomon (Greek Solomōn). The giveaway is that Greek had no sh sound.

If anything, Protestantism has reintroduced some Hebrew names in the King James Version (and the subsequent traditions), that had been Grecisised/Latinised in Catholicism. Compare the names of books in the Douay–Rheims Bible with those in the King James. Isaiah and Zephaniah are not Greek. Nor is Joshua—although the fact that the Greek for Joshua is Iēsous may have had something to do with that too.

How do Greeks distinguish Jesus from Joshua then? By calling the latter “Jesus, son of Naue”—Naue being the Septuagint version of Nun (biblical figure) .