Here's a hard one. Can't find it in the DT. Not surprising as there's only one known oral version collected by Gardiner from James Rampton, Whitchurch, Hampshire, 1906. Published in The Wanton Seed and even then Frank Purslow has had to augment it from broadsides. There are plenty of broadsides starting around 1800 with the 'Young Johnston/Johnson' title. However what appears to be an earlier version printed by Hurd of Shaftesbury, Dorset, has for title 'Young C------___, Or, a warning to Young men.' The arrangement of the stanzas makes better sense.

I'm convinced it's a genuine 'execution' ballad. It fulfils all the stylistic criteria and it has the ring of truth about it. The lad was apparently well-off and well-liked by wealthy ladies who offered £500 to reprieve him. He was convicted of forging a bill (banknote)and was hanged which was the usual punishment regardless of status upto 1830.
I would guess the case was quite high profile. The change of name to Johnson could have come about when a few years later another case came up and the broadside printers wouldn't waste a good ballad. The Johnson version seems to have been much more popular.

Come all young Men of Learning good
A Warning take by me,
Keep your Hands from Pen and Paper
That is call'd Forgery:
It was my ready Wit and Learning good
brought me unto this Place,
And I stood trembling at the Bar
To all my Friends' Disgrace.

My Name it is young C------___,
How cruel is my Fate,
Neither Lands nor Livings can save me
My Crime it is so great;
For the Ladies all around me stand,
Five Hundred Pounds they'd give,
All for the life of C------___,
If they would him reprieve.

Then up spoke one of the Grand Jury--
Saying, Ladies that ne'er can be,
For if you'd give Ten Thousand Pounds,
We never could set him free;
For we have his signing lying by,
As will show in the forged Bill,
And we must hang young C------___,
Tho' it's all against our Will.

Then C------___'s Trial being over--
he was condemn'd to die,
Which caused many aching hearts,
And many a wat'ry Eye;
I am cut off all in my Bloom,
Hard is my Fate, cry'd he,
For the Want of Wit and Conduct
Has prov'd my Misery.

As C------___ went up the Scaffold--
So boldly up spoke he,
So freely I forgive this World--
If this World will forgive me;
And with a placid Countenance
He made a gracious Bow--
Saying--Farewell, farewell, Companions all,
This World I bid adieu.

If anyone has better research facilities it would be great to pinpoint the origins of this one. I tried Googling and came up with several candidates whose names begin with C who were convicted and sentenced to death for forgery of a banknote c1790-1830 and if I could get onto the new trials website I might be able to narrow it down.