Okay... so Jewish New Yorkers in the 50s knew. Nu?
I didn't know. I would never have known she wasn't saying "nue" again there, or "new" or "knew"... all homonyms and words we use constantly in this language (the language these shows are written in). In this song it makes little difference if you don't know - it's either a fun tag if you know or a cute but maybe confusing tag if you don't --- the rest of the song doesn't rely on audiences or listeners knowing the yiddish word "nu".
Unlike "Sue Me", in the case of "Second Hand Rose" it doesn't actually make sense if the word you think you hear is "new" (or "knew" for that matter), but unlike "Sue Me" it doesn't really matter if you get it or if it makes sense or not.