I think the Mozart piece is called:

    Ah! Vous dirai je maman.

There may be an apostrophe between the 'dirai' and 'je'...  I'm sure
our francophone readers will correct me.

I suspect that Mozart derived his tune from a folk melody rather than
the other way around.

The tune was also used by Dohnanyi (I think) in an amusing work for piano
and orchestra whose title escapes me completely.  After a huge and weighty
orchestral introduction the piano enters with a childishly simple version
of the tune.

Jeffrey Borinsky
Oxford, UK