No national has ever had a registry of Jews, though some local communities might have had a list (probably to show who had full rights in the community, and who didn’t).
While some countries have national registers that indicate ethnicity or religion, none of these were based on halacha. In effect, the zionists use a similar system: anyone whose arguably believable claim to have Jewish ancestry and does not claim to be a member of any religion other than some from of Judaism, is considered Jewish for immigration purposes; to the Israelis, it is not relevant that the the individual is an atheist from a long line of atheists and the mother and maternal grandmother were not Jewish.