We maintain that Gentiles can become prosolyte Jews within Nazarene Judaism just as within Rabbinic Judaism.
I can share my personal experience from two months of attending a couple of different Messianic congregations.
It seems to me as though Messianics embrace the concept that the church replaces Israel. So Messianic Jews may themselves embrace the Mosaic covenant but they generally view Christians aka gentiles who do not embrace covenant as equally "saved". This is because what is important to them is that they become "one new man" in the church. So Jews stay Jews and Gentiles stay Gentiles. They also view Jews and others who do not "embrace Christ" as not saved and doomed to eternal destruction in Hell (this is Hellenistic Greek philosophy).
Nazarene theology is different and distinct from Messianics (who embrace Evangelical theology). Nazarenes believe that through Messiah Yeshua covenant is offered to Gentiles as well as Jews. Both must come under covenant to have a relationship with the father, YHWH. Judaism is based on a covenantal relationship with YHWH.
The Mainline of Messianic theology embraces "One Faith Two Expression Theology", the idea that Messianic Judaism and Gentile Christianity are different cultural expressions of the one true faith. However they would NOT identify gentile Christians as Israel, as their theology is more of an adaptation of Church/Israel Dichotomy (as in Dispensationalism) rather than Church Israel Replacement theology.
who is a Jew
he/she whose father &/or mother is a jew.
he/she who is duly converted into the covenant.
he/she who has no other place to go.