Judaism has a lengthy and coherent Scripture. It begins with an origin story, proceeds to doctrine, teachings, hymns, and prophecies, contains well over a thousand years of recorded history, and is laden with what Christians acknowledge to be prophecy.
Christianity traces itself from the birth of Jesus the Christ (Messiah), incorporating many dozens of textual references to the Jewish Scripture. It differs from all other religious instruction, both prior-to and since, in emphasizing a GOD who loves and redeems via blood sacrifice – initially under the Jewish system of temple sacrifices using spotless animals, and later by crucifying Christ, the ‘spotless’ perfect Sacrificial Lamb. He suffered death so that the entire world could be redeemed. The Christian New Testament accreted during the second half of the first Century CE, from the hands of at least eight men. My count isn’t handy, but we have many letters of Paul, gospels from Matthew, Mark, Luke, and John, plus James, Peter, and one or two others whose works are included but not voluminous. They wrote or dictated to scribes a total of twenty-seven books over the span of about fifty years.
Islam arose in a span of twenty years via rhymed couplets memorized by devotees because the originator, and presumably many devotees who committed them to memory, were illiterate. After perhaps as little as a century these utterances were committed to ink, collected but not in the order of their delivery, and used as the basis for a faith system which alleged to ensue from, and correct, both Jewish and Christian teachings. As to any visible connection to either the Old Testament considered sacred by both Jews and Christians and the New Testament held sacred by Christians, the Muslim Scripture has at best a vaporous connection marked almost entirely by its divergences from them.
The Muslim Scripture ‘corrects’ those earlier Scriptures by e.g. placing Moses and Elijah (I think – at any rate, a prophet who lived a thousand years after Moses) as co-contemporaries of Pharaoh and pointedly ‘corrects’ or disagrees with most of the major tenets of both pre-existing Scriptures.
Just for a minor example, the early Muslim Scriptures, delivered during the first ten of their prophet’s twenty active years, seldom if ever refer to their god as Allah – but during the second decade the name Allah, which by the way was the same name used for the Moon Goddess by ‘pagan’ Arabs in that extant culture – As I mentioned, the name Allah found usage primarily in that second decade, to the point where the Muslim writings ‘correct’ the Jewish and Christian writings by chiding them for using their own name for not being ‘Allah.’
For one, the Jewish name was considered too holy to pronounce – so the vowels from one word and consonants from another desensitize it as “Jehovah” – and when translated to e.g. English that holy name is “I AM WHO I AM.” In short, this is the name for the self-created source of the entire physical universe. Yet Muslims take it as a matter of faith that the original and correct name of GOD was that borrowed nomen appropriated after the fact from their pagan moon deity.
So, did the Muslims actually “part ways” with Jews and Christians?
Christianity accepts Jesus as the fulfillment of Jewish prophecies as taken verbatim from the Jewish Scripture. Christians read it today, devoid of the changes or “corrections” which the Muslim scripture alleges to be necessary.
In my unstudied opinion, formed by reading layman-level explanatory works from career-level scholars, the Muslim Scripture hijacks a name here and an idea there, while diverging wholesale from any actual religious teaching found in either the Jewish or Christian scriptures.