The Gatty collection has the following;

The Parson with the Wooden Leg [Roud 1508]
from - 'Down Yorkshire Lanes' Hallamshire Traditions 2013
http://www.hallamtrads.co.uk
Tune coll. Braithwell, S. Yorkshire 1907 R.A. Gatty

A barber there was named Timothy Briggs
Quite famous was he for a-making good wigs
Until with a lass call'd Rebecca Bell
Slap over his heels in love he fell
        Singing fol-the-rol-day,
        Right fol-the-day, right fol-the-day
        Right fol-the-laddy,
        Right fol-the-day

The went to the church the knot for to tie
To a wooden-legg'd parson named Jonathan Sly
And sure, if you'd seen him you'd laughed at him plump
As he mounted the pulpit with his old stump

They'd only been married a week or two
When Rebecca turned out a most terrible shrew
No comfort have I with this woman he said
I'll go back to the parson and I'll get unwed

He went to the parson, said he, 'Mr Sly'
If I live with this woman I surely shall die
You said you would make us two into one
I've come to see now if I can't be undone

The parson he said, 'That's a thing rather new'
I don't know that I'e power my flock to undo
Here's hoping you'll lead a more happier life
I'll go round to your house and I'll lecture your wife.

Now the barber, quite pleased went taking his glass
The old parson stumped off to lecture the lass
But when he got home, Lor', what did he see?
But the parson with Rebecca on top of his knee

Now the barber bristled up every hair
He said, Mr Parson what are you doing there?
You said you wanted undoing my man,
So you see I'm a doing it fast as I can

I think I'm undone now if I never was before
He kicked Mr Parson straight out through the door
He lay in the street and his wooden leg stood
Like a spade sticking up in a cartload of mud

This couple from then they lived more reconciled
But nine months from that day she brought him a child
The barber hung himself up on a peg
When he found his child born with a brand-new wooden leg.

The following, while not strictly about a barber, has the distinction of being the only time a tune was collected with this text:

A Nobby Head of Hair       [Roud 12612] Source as above.

You've call'd on me to sing a song,
I'll try what I can do
I don't say whether good or bad
For that I'll leave to you
The subject's now before you
And I solemnly declare
There's no-one in this street can sport
Such a nobby head of hair

When an infant, I a wonder was
But upwards as i grew
At school I so surprised the boys
They in mobs around me flew
But when a young man I had grown
My mother said, if I took care
I soon should catch an heiress
With my nobby head of hair

Although my hair is elegant
It oft gets into scrapes
At the Zoo, the other day
Twas well pulled by the apes
And making my escape from them
I was grappled by a bear
It fancied that I was its cub,
With my nobby head of hair

Not liking this brute treatment
From the gardens I did roam
I caught a lady ogling me
I asked to see her home
Her husband coming up the road
He asunder did us tear
Then he dragged me through a horse-pond
By my nobby head of hair.

He left me near dead with affright
And wet through to the skin
A mob soon came around me
They did naught but jeer and grin
A p'liceman took me in custody
And solemnly did swear
That I one of the swell mob was
By my nobby head of hair

To the magistrate my innocence
I pleaded, but in vain
He said to prison you mus go
Your guilt it is quite plain
So to the treadmill I was sent
On the silent system there
But what grieved me most was thy cut off
My nobby head of hair

But now that I am free again
I'm happy as a king
Thats one reason why tonight you see
I have come here to sing
But here's a fact you can't deny
It is a thing most rare
To see a handsome chap like me
With such a nobby head of hair