This site says that "Piddlin Pete" was written by Leslie Sarony, who is also the source of Ain't It Grand, To Be Bloomin Well dead.
-Joe Offer-

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PIDDLIN' PETE
(Leslie Sarony)

A famous dog once came to town
Known to his friends as Pete
His pedigree was ten yards long
His looks were hard to beat

And as he trotted down the road
'twas beautiful to see
His work at every corner
Every post and every tree

He never missed a land mark
He never missed a post
For piddling was his masterpiece
And piddling pleased him most

The city dogs stood looking on
In deep and jealous rage
To see this little country dog
The piddler of his age

They smelt his efforts one by one
They smelt him two by two
But noble Pete in high disdain
Stood still 'til they were through

Then when they'd smelt him everywhere
The praise for him ran high
But when one smelt him underneath
Pete piddled in his eye

Just then to show these city dogs
He didn't care a damn
He strolled into the grocers shop
And piddled on the ham

He piddled on the cornflakes
He piddled on the floor
And when the grocer threw him out
He piddled up the door

Behind him all the city dogs
Debated what to do
They'd hold a piddling carnival
The hoop they'd put him through

They showed him all the piddling posts
They knew about the town
And off they set with many a wink
To wear the stranger down

But Pete was with them all the way
With vigour and with vim
A thousand piddles more or less
Were all the same to him

And on and on went noble Pete
As tireless as a windmill
And very soon those city dogs
Were piddled to a standstill

Then Pete an exhibition gave
Of all the ways to piddle
With double drips and fancy flips
And now and then a dribble

The city dogs said farewell Pete
Your piddling did defeat us
But no one ever put them wise
That Pete... he had diabetes.

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Also see this page (click).