"Hey nonny nonny" is a common nonsense line in madrigals, from the English Renaissance. There is a song in Shakespeare's "Much Ado About Nothing" that uses it thus:
Then sigh not so
But let them go
And be you blithe and bonny
Converting all
Your songs of woe
To "Hey Nonny Nonny"
The idea here is to turn your sad sighs into a happy, silly song that doesn't mean anything.
"Ha cha cha" is a set of nonsense syllables that was used about 350 years later in American Jazz songs. There are a number of songs from that period that go further and put the two lines together for comic contrast. The "Hey Nonny Nonny" reminds listeners of boring old madrigals, and the "Ha cha cha" brings them smack into the latest slang of the time.
The first example I found of this expression is in a song from the musical "Of Thee I Sing", in 1931, by George and Ira Gershwin. The chorus sings "With a hey nonny nonny and a ha-cha-cha" when they are jubilantly celebrating the announcement that the President is going to have a baby.
I also remember hearing the line in the Marx Brothers movie "Duck Soup" from 1933, They make a mockery of big musical numbers when their country decides to go to war, and they bring in all kinds of silly cliches and lines from other songs of the time. This is one of the lines they steal.
The most popular film to use the line is probably "Singin' in the Rain", from 1952. There it appears in the song "Fit as a Fiddle", which is supposed to be a rowdy "old" number from the vaudeville days. Again, it is used as a silly expression of fun, and also to remind the audience of an earlier time (although the vaudeville show would have been even earlier than the 1931 appearance in "Of Thee I Sing".)
My conclusion is that the line doesn't really "mean" anything, but it is commonly used in fun songs to fill out a phrase with happy nonsense.