It's not on any of my 3 Rough Deal String Band CDs. Found two sets of lyrics, maybe not what you are looking for, both have references to July in them. Parts of both are similar to Rough Deal's "Jack of Diamonds" but there's no mention of a cuckoo in that song.

THE CUCKOO
Peter, Paul & Mary

Oh the cuckoo she's a pretty bird, she sings as she flies
She never gets lonesome till the first day of July.
I've gambled in England, and I've gambled down in Spain.
I gambled with five aces, now I've gambled my last game.

Oh, it's gamblin' that's brought me prison,
And it's gamblin' that's brought me pain
I'll never see the cuckoo or hear her song again

Jack o' diamonds, jack o' diamonds, I know you of old
You robbed my poor pockets of silver and of gold.

Oh the cuckoo she's a pretty bird, she sings as she flies
She never gets lonesome till the first day of July.

www.sing365.com

Cuckoo Lyrics
Kristin Hersh

Oh the cuckoo she's a pretty bird
She wobbles when she flies
She don't ever hire a cuckoo
Till the fourth day of July
Jack of diamonds
Jack of diamonds
I know you of old
You rob my poor pockets of silver and gold

Oh the cuckoo she's a pretty bird
I wish that shh was mine
She don't ever drink water
She only drinks wine

Gonna build me a log cabin
On a mountain so high
So I can see Willy
When he goes on by

Oh the cuckoo she's a pretty bird
She wobbles when she flies
She don't ever hire a cuckoo
Till the fourth day of July

www.lyricsdepot.com