I found several versions of this on the Internet. I get the impression that they were all transcribed from audio by people who didn't know any Italian. Maybe the "Italian" parts are nonsense anyway, but I have a hunch a real Italian speaker would have spelled them differently. Is there an Italian-speaker out there who can clarify?

Lyrics copied from http://kokomo.ca/early_years/chi_baba_chi_baba_lyrics.htm

CHI-BABA CHI-BABA (MY BAMBINO GO TO SLEEP)
(Music, Al Hoffman & Jerry Livingston. Lyrics, Mack David. 1947.)

Many a year ago in old Sorrento,
A certain ditty was quite the thing.
Whenever a mother rocked her baby in Sorrento,
This little ditty she used to sing:

Chi-baba, chi-baba, chi-wawa
An' chi-lawa kook-a la goombah
Chi-baba, chi-baba, chi-wawa
My bambino, go to sleep!

Chi-baba, chi-baba, chi-wawa
An' chi-lawa kook-a la goombah
Chi-baba, chi-baba, chi-wawa
My bambino, go to sleep!

All the stars are in the skies ready to say "goodnight."
Can't you see your doll is sleepy, too?
Close your drowsy little eyes. Mama will hold you tight
While she sings a lullaby to you:

Oh, chi-baba, chi-baba, chi-wawa
An' chi-lawa kook-a la goombah
Chi-baba, chi-baba, chi-wawa
My bambino, go to sleep!

(Chi-baba, chi-baba, chi-baba, chi-baba, chi-wawa, and chi-lawa)
Chi-baba, chi-baba, chi-wawa
An' chi-lawa kook-a la goombah (la goombah!)
Chi-baba, chi-baba, chi-wawa
My bambino, go to sleep!

(Chi, chi, chi-baba!)
Chi, chi, chi-baba, chi-baba, chi-wawa
An' chi-lawa kook-a la goombah (la goombah!)
Chi-baba, chi-baba, chi-wawa
My bambino, go to sleep!

All the stars are in the skies ready to say "goodnight, goodnight."
Can't you see your doll is sleepy, too?
Close your drowsy little eyes. Mama will hold you tight
While she sings a lullaby to you, ever so sweetly!

Chi-baba, chi-baba, chi-wawa
An' chi-lawa kook-a la goombah (la la goombah! chi-baba!)
Chi-baba, chi-baba, chi-wawa
My bambino, go to sleep!

Chi-baba, chi-baba, chi-wawa
My bambino, go to sleep!
Chi-baba, chi-baba, chi-wawa
My bambino . . . Go to sleep!

[Recorded by Perry Como, 1947.]