Chuckle, I like this question. There is a trick to it.
The above answers are correct:
Constanze was Mozart's wife's name and thus her nickname is Stanzi.
Mozart's first name was Wolfgang, so it is shortened in the movie to Wolfie.
The trick is the question of the son's nickname. Where the trick lies is that the nickname if you are not paying attention sounds as if the son was named after his father. The son in question is Franz Xavier "Wolfgang" Mozart the younger of the two living sons. He was baptized Franz Xavier Mozart; however, his whole life he was called Wolfgang by his parents. This is historically accurate.
This is a particularly difficult question because unless you are really paying attention, you make the assumption that Wolfgang is the boy's name. It is also not as cutesy as his mother's and father's.
Interesting side note: the only officially recognized nickname in Mozart's immediate family (aside from his son), was his sister: Maria Anna Mozart nicknamed: Nannerl.
Mozart is called Wolfie, Canstanze is stanzi. I scanned throught the script and I think their son Karl was called "monkey."
There weren't any nicknames, I believe. WOLFIE is just a shorter version of Mozart's first name, WOLFGANG. In the same way, STANZI is a reduction of his wife's first name, CONSTANZE. By the way, "Amadeus" is Mozart's middlename.