Successor is not the appropriate word for the religion of Christianity in relation to Judaism.
Christianity is a replacement theology. It has much more in common with the beliefs of the Greeks and Romans, and attempt to obliterate the eternal covenant and remove any self-determination of the Jewish people.
At best, Christianity is a little less than 1/3 based on Jewish precept, but primarily a mixture of the beliefs of the polytheistic peoples in and round Judea and the Galilee of the first century. Only Judaism had strict prohibitions of altering or adding to the Torah or the concept of God and the path of the eternal covenant of Israel.
Ancient conquered peoples and polythestic nations were accustomed to their political leaders telling them what gods were to be venerated during their rule and which deity their ruler was representative of in human form. Adding a new deity or giving a new name to an ancient deity whose belief was already established was not that big a deal for most of the non-Jewish world. Tanakh recorded that any time such a practice of a Jewish king telling the Jewish people that they were to worship a foreign deity...the entire Jewish people suffered..and often at the very hands of the people whose deity they had left God to serve.
Every aspect of Jewish belief is given a new spin in the New Testament doctrine. Things that I once thought must have been fabricated outright I later discovered were beliefs of the ancient Greeks and Romans. This was the longstanding battle for souls going on..the Hellenized Jews, those who had become apostate to Judaism through four centuries of Greek rule, desired to entice other Jews to worship as the Greeks that they believed superior in philosophy and knowledge, so they created texts that tried to syncretize beliefs. However, reality was that those beliefs were identical to the beliefs and practices that the Torah forbids to the Jewish people to keep them separate as a covenant nation dedicated to God alone. Then we had Roman occupation to follow the Greeks.
Once one begins an in-depth study of the religious practices and beliefs of all the peoples surrounding the Jews in the centuries just preceding the beginning of Christianity, one can see these developments take place. It was a concerted effort to do away with Israel and Judaism. Politics and religion, rule and belief were inseparable in ancient times in the Levant.
As to your additional detail about Islam. It is a replacement theology that came along after both Judaism and Christianity, so that's why it claims the other two older religions are corrupt. It was an attempt to syncretize both previous religions and then also became a tool of conquest.
Islam returned away from the incarnate deity of the New Testament that so egregiously violates Torah precept, but it also has elements of belief that are equally incompatible to Torah precept. Neither religion is a successor to Judaism, but a replacement attempt.
The first to follow the New Testament doctrine would have been apostate Hellenized Jews and Romans..the doctrine is not Judaism. The Christian concept of "Messiah" in the New Testament is not a notion that supports Judaism, but a concept of assimilated Jews and Roman beliefs that is a clear attempt to negate Judaism.
Hopes of a KING, an anointed human ruler who will fulfill a prophetic vision to break the yolk of religious and political persecution FIRST for the Jews, then for the rest of humanity ..is the Jewish messianic hope for the prophetic vision of a Davidic messiah. His rule will bring about peace on earth for all humanity and all will know God without being taught. There was no prophecy that it would be God incarnate as man in violation of direct Torah commandments. There were MANY messiahs in the Hebrew Bible. Messiah means anointed ruler. Even a NON Jewish King, the Persian King Cyrus is called a messiah directly in the Tanakh, for his righteous rule. Christianity is replacement theology to Judaism and its holy texts and precepts depend on turning the beliefs of Judaism topsy-turvy in many regards. It is so very different from the concept of the nature of God to the manner that humans relate to God and to each other.
Any entity that would threaten Roman authority would be persecuted by the Romans. Early Christians ( a band of followers of a messiah hopeful would threaten Roman rule, since the prophetic vision of Judaism of "the messiah" is of a RULER, a king, who breaks the yolk of political oppression and persecution for all and brings about universal brotherhood, peace and knowledge of God) would most certainly have been a threat. But once the doctrine of the New Testament gelled, that changed the Tanakh concept of messiah meaning an anointed king, to mean a sacrifice for sin and a "kingdom" not of this world, this had effectively changed the Christians from being enemies of Rome to tools of Rome.
The Temple treasury was raided in 66CE by the Roman governor of Judea, Florius and a massacre of Jews in Jerusalem took place. Christians did not take part in the subsequent revolt and were thus rewarded by the emperor Nerva (96-98 C.E.) who freed the Christians from paying the Fiscus Judaicus, the Jewish capitation tax decreed as a punishment for the revolt of 66-73 C.E.
Clearly, the Romans already regarded the Christians as a separate religion group. Decades later, during the Second Jewish revolt, 2nd revolt, once again Christians refused to ally themselves with the Jews and allied themselves to the Romans. This led to another messiah hopeful, Simeon Bar Kochba to view them as the enemies of Israel, thus, after the defeat of the Jews and the murder of Bar Kochba, the Chistians were not expelled along with the Jews.
After Constantine - First Christian Emperor of Rome (285-337 CE) the Roman empire was Christian.
edit: To Julia's answer. Some major mistakes there, who the messiah is, isn't the biggest difference/incompatibility with Judaism and Christianity. The affirmation of faith and purpose of the eternal covenant nation people to be a nation dedicated to the incorporeal Creator alone forbid Israel from worship of any incarnate deity/man as a god. Any Jew who has ever or ever will abandon Judaism/faith in God alone and then rely upon Jesus as either an incarnate manifestation of a god or as a sacrifice for sin has willingly separated themselves from the covenant nation people, Israel and their Jewish heritage and observances of Torah according TO the Torah. Repentance and return are always open to them.
Worship of Jesus or through him or reliance upon him for atonement are egregious violations of Torah for any Jew. That is Christian doctrine of their "New" Covenant. It remains incompatible with the eternal covenant of Torah.
The righteous of all nations merit blessing. God is never exclusive to the Jews, but the covenant of Israel demands that the Jews worship God EXCLUSIVELY. No believing Jew worships a man as an incarnate deity.
Of course there are people who are raised in every religion who decide to leave it for another when they grow older. That's their choice.
The biggest misconception around is that Judaism is like Christianity without Jesus or to say Christianity is like Judaism except it has Jesus as the Messiah.
The Christian concepts of many terms used by both religions do not mean the same things as they do to the Jewish religion.
The two religions are so very different on so many levels that it is simply impossible to adequately address them all under this question but I can give you a good starting point to learn the differences.
Christianity focuses on teaching that one must accept on faith belief that Jesus death and blood exclusively "reconcile" a sinful nature of humans to God and then they will be given an awareness of God that will make them a new person, impart eternal life and change their behavior. B eing born with a burden of sin is foreign to the Torah that teaches that humans are born with propensities for both good and evil and have the ability to master our evil inclinations. That ability is noted the very first time the word sin ever appears in the Jewish Bible. Judaism teaches us that it is how we live and interact with one another that opens us up to the awareness and direct connection to the Divine. For many people, it appears to be a slow process of growth into an awareness of our connection to God, rather than some kind of overwhelming divine revelatory moment. The entire nation of Israel had direct Divine revelation at the Exodus and at Sinai when Israel rededicated itself as a holy nation dedicated to God. Judaism is a path of living our life from God.
Christianity is focused on stressing the ultimate importance of what happens after death and that unless they believe a certain way, they merit torture rather than eternal life in heaven.
Shalom