I am the only member of my immediate family who isn't Catholic. And the rest are seriously Catholic (they go to church every sunday and holy day of obligation, two of them went to Catholic universities, my brother is the only fundamentalist Catholic I've met, and my father converted despite being fostered intellectually by a borderline atheist German Jew). So while I don't keep up with what the local archdiocese is up to, or even what the pope is, I do know something of the structure of authority in the Catholic faith. It is not an "oral law". The pope's word is not usually that of god, and there is a specific and little used formalized ritual for when the pope speaks for god. As for most Christian sects, they are defined by conceptions like sola scriptura, where the preacher can say whatever she or he wishes, but what matters is the bible. It's been almost 1,500 years since the Talmudim were written. Rabbis have, of course, continuously studied the Torah and rabbinic commentary, but it is not oral law. As for aḥādīth, these are already categorized into very specific groupings. Also, Ibn Shaq's life of Muhammad (PBUH) was never oral but has shaped Islamic understanding for centuries.
It seems as if you are applying a 1st and 2nd Jewish concept to religions in general or at least those related to Judaism. But I'm not sure how applicable this is. =
I can't speak for myself, but I know that at least officially the Roman Catholic church has the "authority" to speak for god, while the bible is only inspired by god.