From the sheet music at the The National Library of Australia:


THE WOODPECKER SONG / REGINELLA CAMPAGNOLA
English words by Harold Adamson; original Italian words by Bruno Cherubini (pseudonym "C. Bruno"); music by Eldo Di Lazzaro; ©1939.

He's up each morning bright and early
To wake up all the neighbourhood,
To bring to ev'ry boy and girlie
His happy serenade on wood.
Hear him pickin' out a melody:
Peck, peck, peckin' at the same old tree.
He's as happy as a bumblebee,
All day long.

To serenade your lady,
Just find a tree that's shady,
And when you hear that tick-a-tick-tick, tick-a-tick-tick, tick-a-tick-tick, sing right along.
Come on and try his rhythm
And let your hearts beat with 'im.
Just listen to that tick-a-tick-tick, tick-a-tick-tick, happy little woodpecker song.

[There is a Wikipedia article about this song.]