It's not nanny who's sick at heart! (5)
I believe the answer is:
billy
'it's not nanny' is the definition.
I can't tell whether this definition defines the answer.
'who's sick at heart' is the wordplay.
'who's' becomes 'by' (I can't explain this - if you can you should believe this answer much more).
'sick' becomes 'ill' (synonyms).
'at heart' is an insertion indicator (inserted letters go to the heart of the word).
'by' going around 'ill' is 'BILLY'.
Can you help me to learn more?
(Other definitions for billy that I've seen before include "Goat; camper's kettle" , "Cooking can for William" , "The familiar male goat" , "Light metal cooking pot used for camping" , "... boy" .)