To answer this question, we need to take a brief survey of the Jewish faith. Strictly speaking, you don’t have a Jewish faith until the giving of the Law to Moses on Mount Sinai in Exodus 19:1ff. Until that point, you have had the Hebrew people, or the people of Israel, but the official establishment of the descendants of Abraham, Isaac, and Jacob into the nation of Israel as a political and religious entity occurs during the period of the exodus.

After that, the Law of God (or Mosaic Law) guided the people of Israel in their civic and religious duties well into the monarchic period. While the period of the judges saw cycles of sin and idolatry and the history of the divided kingdom post-Solomon saw even greater periods of idolatry and apostasy, there have always been a faithful remnant of Jewish people who held firm to the religion passed down from God to Moses upon Sinai.

That takes us from roughly 2100BC to 586BC. In 586BC, the northern kingdom of Judah was conquered by the Babylonians and many Jews were taken into exile in Babylon (the southern kingdom having fallen about 136 years earlier in 722BC). For a period of 70 years, there was no proper practice of the Jewish faith. That is until approximately 516BC when the exiles began returning to Jerusalem. The period of 516BC to 70AD marks the time known as Second Temple Judaism. A new temple had been built (later expanded by Herod the Great), and many Jewish practices we see in the in the NT were established during this period.

Ezra, who was a priest and a scribe, introduced the practice of the public reading of the Scriptures and the explanation of the Torah into Jewish religious practice (a practice that is still followed today). We also see the rise of the synagogue during this period as well as the different sects of Judaism that were prevalent during the “inter-testamentary” period — namely, the Pharisees, the Sadducees, the Essenes, and the Zealots. The Pharisees were theological conservatives, the Sadducees were theological liberals, the Essenes were the separatists who lived in secluded communes, and the Zealots were the radicals who took violent action on the Roman occupation.

The recognition of Jewish oral law as important led to the Pharisees becoming increasingly important to Jewish religious life as they took the oral law and created many practices that defined the boundaries of proper Jewish life. It was also during this period that the Septuagint was published. The Septuagint (or LXX for short) is a Greek translation of the OT Hebrew Scriptures. Much of what we see during Second Temple Judaism shapes Jewish life going forward. Also, Christianity began in this period as well as being an outgrowth of Second Temple Judaism.

Now in 70AD something special and cataclysmic happened that affected Judaism for centuries to come — namely, the destruction of the temple and the sacking of Jerusalem by the Roman armies of Titus Vespasian. Aside from the “Nazarenes” (i.e., Christians), the only other Jewish sect to survive were the Pharisees. That means what we see as modern day Judaism evolved from sect of the Pharisees from this period of time. From 70AD to 1750AD, the rise of Rabbinic teaching grew into prominence. Great Jewish teachers developed followings that shaped faith and practice for Jews going forward. Thus the development of the Talmud, a compilation of rabbinic teachings that reached its final written form around 500AD.

One other significant change for Judaism was the transition from physical animal sacrifice in the temple to other rituals and symbolic reminders, liturgical references, and spiritual exercises such as prayer, repentance, and good deeds. With no temple, and Jerusalem crawling with Roman soldiers, the Jews had to forego temple worship for these substitutionary measures.

As it stands today, there are three major strands of Jewish practice (these are broad groupings):

1. Orthodox Judaism — Very conservative, believes in the Torah as divinely revealed, and strictly observes Jewish laws and customs
2. Reformed Judaism — Unlike reformed Christianity, very liberal beliefs. Does not believe in the divine inspiration of the Torah, and takes a very humanist approach to religious traditions.
3. Conservative Judaism — Somewhat of a “half-way house” between the Orthodox and the Reformed. Follows the Torah, but does not believe that the Torah is “frozen in time,” but to be re-interpreted for modern times. Revelation is/can be an ongoing process.

One can probably add to these three broad groups two more: (1) Hasidic Jews, and (2) Messianic Jews. The Hasidic Jews are considered to be somewhat “mystical” in their approach to religious practice. While the Messianic Jews are those who recognize Jesus as the Jewish Messiah, but still retain their Jewish identity.

So to answer the question as to when did the Jewish faith become the modern Jewish religion, it would be in that time period between the destruction of the temple in 70AD and today. That’s when the foundation for modern Judaism was formed. But in regards to the part of the question that says “the Jewish religion that doesn’t believe in Christ as the Savior or Son of God,” there has always been those in Judaism from any period who have rejected God’s revelation.

The OT sacrificial system was always typical and symbolic pointing to the true substance that can be found in Christ. Christ is the fulfillment of the Jewish ceremonial law. That’s why the temple was destroyed in 70AD — it was a judgment on the Jewish people for rejecting their Messiah. While the Romans were the proximate cause, it was God as the ultimate cause of the destruction. It was His way of saying OT religious worship is a relic of the old age. Jesus Christ, by His death and resurrection, made the old covenant obsolete. It was replaced by the new covenant and a better Mediator, Jesus Christ the Righteous One.