What does 才 mean in this sentence?
(Quote taken from a nursery rhyme.)
After looking in the dictionary my best guess is that it means "just" or "only", so as to make the sentence mean "Mother says only a long nose is beautiful."
I think a good way to translate that sentence into English so that the usage in Chinese is well understood would be to say "Mom says that if your nose is long, only then is it pretty". I think it's important to think of 才 in this case meaning "only then" rather than just "only". That makes a sentence like 你跟我去我才去 easier to understand ("I'll only go if you go with me" or "If you go with me, I'll go" or "You go with me, only then will I go").
Be careful not to confuse this meaning of 才 with its meaning in a sentence like 我才不去 in which 才不 means certainly won't or certainly do not.
Yes, your interpretation is correct. So the mom indicates that a short nose is not beautiful by saying "鼻子长才漂亮".