At the beginning of the twenty-first century, Jewish identity remains a complex matter. It is an ideological and halachic issue that often divides Jews from one another. The question you are asking, then, cannot be answered with absolute certainty because religious ‘movements’ and even Jewish organizations have different policies with regard to membership and affiliation.
Traditionally, Jewish identity is defined by one’s mother. That is, if one has a Jewish mother, then one is considered Jewish according to Conservative and Orthodox rabbis; if one’s father is Jewish but ones’ mother is not, then one would not be considered Jewish without undergoing a formal ceremony conversion, ‘according to Jewish law.’ This would involve a visit to the mikveh, a ritual pool for both men and women, and either circumcision or a ceremony called hatafat dam brit, for men.
The Reform and Reconstructionist movements embraced a policy in favor of patrilineal decent so that any person who has a Jewish parent and identifies himself or herself as Jewish would be considered Jewish.
As a Conservative Rabbi, I often encounter people who have non-Jewish mothers and Jewish fathers. While conversion is necessary in such cases, I usually treat such conversions differently than I would a person who seeks to convert to Judaism without any prior Jewish lineage. This is particularly the case when the person in question has been raised as a Jew without a formal conversion. In such cases, I explain that they while they are Jewish in identity, they need to formalize their identity and practice in law, much as a person born in a foreign country but raised in the United States may feel American but not have the benefit of citizenship because they were not formally naturalized.
Jewish organizations may vary in terms of their policy on Jewish identity. National and international organizations that are synagogue based may have the same requirements as the synagogue. Many other organizations, because they have a non-religious in nature, do not question the basis of one’s Jewish identity. Simple identification is enough. This would not be the case for an Orthodox or a Conservative synagogue. In such cases, a conversation with the rabbi of the congregation will help clarify what is expected of members.
Following the ritual of conversion, you would be considered Jewish and there should be no barriers to your full participation in the synagogue and its various organizations. In fact, the Talmud prohibits reminding a person of their prior status. The person who undergoes conversion is considered no different than someone who is born to two Jewish parents.
Your acceptance or non-acceptance as a Jew by an organization or a synagogue is not a judgment of you as a person. It is recognition that we live in a complex world where we don’t all agree on what it is that defines Jewish identity. Your passionate sense of Jewish identity should be honored and respected but at the same time, I hope you will be respectful of the need of people to define Jewish identity based on their understanding of Jewish tradition and their conscience.