Remsen can fact-check me here, but I believe being a Jew is a religion, not a nationality. So, one cannot be "part jew".
You can be either. Others have already detailed the conversion process in becoming a Jew as far as religion. Anybody who is a physical descendant of Abraham is some part "Jewish" unless you discount Judaism by birth. Israel is a nation but it is also referred to as a people. ie the descendants of Abraham. Lots of people are "part Jew" and don't even know it.
As to their religion...I have no idea. Being a Christian isn't predicated on what anybody else says or thinks other than God Himself and you, the adherent. Maybe the Jews have a say in who is and who isn't.
