Here are the lyrics, as I learned them.

P STANDS FOR PADDY
Source: Bard (in Canada)

P stands for Paddy, I suppose
J for my love John,
And the W stands for false William
And Johnny is the fairest man
    Johnny is the fairest man, my dear,
    Johnny is the fairest man.
    And I don't care what anybody says,
    But Johnny is the fairest man.

I walked out one May morning
For to take a pleasant walk,
I sat meself down on an old stone wall
To hear two lovers talk.     To hear what they might say, my dear,
    For to hear what they might say
    So that I might know a little more about love
    Before I go away.

P stands for Paddy, I suppose
J for my love John,
And the W stands for false William
And Johnny is the fairest man

"Come and sit you down beside meself,
Over here on the pleasant green,
For it's been three quarters of a year or more
Since together we have been."

"No, I'll not sit by you," says she,
"Not today nor at any other time!
For I hear you love another pretty girl,
And your heart's no longer mine.
    Your heart's no longer mine, my dear,
    No your heart's no longer mine.
    It's been three quarters of a year—no more—
And your heart's no longer mine."

P stands for Paddy, I suppose
J for my love John,
And the W stands for false William
And Johnny is the fairest man

I will climb a tall tall tree
And I'll rob the wild bird's nest.
Then I'll come down with what I find there
To the arms that I like best.
    To the arms that I like best, my dear,
    To the arms that I like best.
    Yes, I'll climb right down and run straight home
    To the arms that I like best.

P stands for Paddy, I suppose
J for my love John,
And the W stands for false William
And Johnny is the fairest man
    Johnny is the fairest man, my dear,
    Johnny is the fairest man—
    And now I know a little more about love
    Before I travel on.