Guest, yes, that is the tune used by Mozart. Speculation that it is Irish, like most great melodies, is wrong in this case. Mozart's set of variations is on the net.
Here is the url. Do I dare to make a link?

www.empire.k12.ca.us/capistrano/Mike/capmusic/form/theme%20and%20/variations/Twinkle/twinkle.htm
Oh well, nothing ventured....
Mozart Variations