This exact question has been posted before! I'm not very happy with my answer, but it covers the basics, at least about circumcision. (Some) Arabs have practised circumcision long before Islam and some scholars would argue that Islam was just as close to Judaism as Christianity was. Beyond that, it is difficult to say anything for certain.

Christianity did not exactly reject the food laws and circumcision. The letters attributed to Paul, particularly the one to the Galatian Christians (in central Asia Minor) expressly state that while Christians of Jewish origin were still observing these practices, they had no right to require Christians of non-Jewish origin to do so. Evidently there were many early Christian converts from Judaism who believed that it was necessary for converts to Christianity from other backgrounds ("gentiles") to follow the Mosaic law, just as it had been necessary for converts to Judaism to become circumcised and subsequently follow the law. This caused controversy as increasing numbers of non-Jews converted to Christianity, and to some extent it was necessary to draw a line in the sand to distinguish converting to Christianity as being something different than converting to Judaism. Since the vast majority of Christian converts were (eventually) of non-Jewish origins, observance of dietary laws and practice of circumcision were just never adopted into Christianity.

I don't know as much about Islam, so I can't say for certain why Islam DID take those practices on board, but certainly the relationship of Islam to Judaism was quite different than the relationship of Christianity to Judaism. Initially, all Christians were converts from Judaism, where this was not the case at all for converts to Islam. I'm sure there are many other reasons why Islam did adopt these practices, but the issue never came up (to my knowledge) as a controversy between different groups of Muslims that could cause dissension in the community of believers.

This question may be too vague to answer, but how much of a role did Hellenistic or Roman influence (political or cultural) play in weakening Christian communities' adherence to Jewish tradition? For instance, were there widespread anti-Semitic tendencies within the Roman Empire that groups of Gentile Christians might have been susceptible to and that might have competed with and eventually won out over any allegiance to the Jewish roots of the movement?

While it would be interesting to consider cultural (Arabs considered their fellow-semites and linguistic neighbors, "cousins") and political (early Islamists were in conflict with the Romans, who were Christians, and wooing a culturally similar and oppressed minority under Rome, was smart politics) angles, if I may add a bit of a theological bent, which was perhaps offered as a justification at that time, to the conversation:

Islam sees itself as the same divine message to all humanity that was sent over and over via different prophets (Jesus/Moses/Abraham being just some the many) and therefore, it recognizes, within limits, the holy books of the Jews and Christians and the practices prescribed therein, with the caveat that the commands of the Quran supersede where the Jews and Christians "corrupted" the divine message.

Also, and perhaps more pertinently, Arabs claim to inherit, via Ismail, the covenant ("Covenant of the Pieces") that God struck with Abraham. Muhammad reclaimed / purified that inheritance with his Revelation and subsequent purification / destruction of the Meccan temple (which, per Islamic theology, was also built by Adam, then restored by Abraham & Isaac).

So, Islam sees itself as a direct continuation of Abrahamic practices and sees Christian deviations as just that.