"The Jews killed Jesus" isn't the earliest form of anti-Judaism-- your second version, about Jews being bad for a nation, is actually older. By around the beginning of the first century BCE, Jews were living all over the Mediterranean, including in cultured, Greek cities like Alexandria. And when Alexandrian Jews wanted to be citizens of Alexandria, the responses from other Alexandrians were mostly about how the Jews' religious practices make them bad citizens who aren't like everyone else and could never be fully loyal to Alexandria.

Keep in mind these are people living under the pre-Christian Roman Empire, so they have no Christ-based reason to hate Jews. So the claim that Jews are bad for a nation goes back much much further than the Nazis, although of course the Nazis had a new and horrible version of it.

The conventional answer is that that sort of shift happened with the enlightenment, in the 19th century. People preferred to view their bigotry in national or scientific terms, rather than religious ones.

However, I prefer to see it as a gradual change, not a sudden one. As /u/TheApiary said, the notion of Jews being a group who can never really be [insert nationality] is quite old. But there's a long gap where it's hard to document much. Still, in the Middle Ages similar notions abound--Jews aren't just Jesus-killers, they're insidiously killing Christian children, or stealing Christian's money with their usury or general monetary unfairness, etc, etc, etc. Anti-Jewish sentiment has never been limited to purely religious phenomena. The change was a de-emphasis of the religious element, and the gradual diminishing of the importance of converting Jews to Christianity (which never was the exclusive goal of it).