Strictly speaking, the "cockney parody", so called, is the original All Around My Hat.  There is no record of any earlier song of that title, or containing those verses.  Of course, there are related antecedents.  To continue:

Further enquiry reveals that the tune in Popular Music of the Olden Time referred to by Kennedy as Chappell's "original" tune for All Around My Hat is in fact nothing of the kind.  My paragraph above concerning The Miller of Dee, therefore, is completely irrelevant to this discussion.  Whence the error in Kennedy I have no idea, but the tune Chappell gave in his earlier collection (as Number 90 in National English Airs) is completely different, and unmistakably a variant of the well-known Hat tune.

Chappell's tune was given by Anne Gilchrist in a note to So Selfish Runs the Hare, a song collected by Cecil Sharp from Mr. Jos. Alcock of Sibford Gower, Warwickshire, in 1922 (Journal of the Folk Song Society, Volume 8, issue 31, 1927), which was sung to another variant of the melody.  Mr. Alcock's text and tune are in the DT:  So Selfish Runs the Hare,  though no source is named.

Ms. Gilchrist also pointed out that  Dashing Away With the Smoothing Iron  and The Poor Little Fisher Boy are also sung to variants of the same tune.

The Journal of the Folk Song Society, no. 34, 1930, has a set of The Nobleman's Wedding noted by H.E.D. Hammond from Mrs. Crawford of West Milton, which includes the All Around My Hat verse and is sung to another variant of the same tune.  A. Martin Freeman commented:

"...the occurrence of the All Round My Hat verse in connection with this tune, which belongs to the same group as the five Nobleman's Wedding tunes in the Complete Petrie Collection, all apparantly variants of the same original, opens up a new and interesting question.  Have we here the lost original of the serio-comic All Round My Hat of the 1830s?  ... In the text of All Round My Hat given in Baring Gould's Garland of Country Song, all that is not new belongs to the comic song, so that the source of the one presumably old verse is still unknown.  But compare the line If anyone should ask me the reason why I wear it [i.e. the willow round his hat] with verses 3 and 4 in Mr. Sharp's Appalachian version of The Nobleman's Wedding (The Awful Wedding):

If anyone should ask the reason
Why I put on my strange attire,
I'm crossed in love, that is the reason,
I've lost my only heart's delight.  [?desire]

But I'll put on my strange attire,
And I will wear it for a week or two,
*         *         *         *  
Till I change my old love for the new.

These verses are part of the song the old lover sang to the bride, and are followed by the half-verse:

But how can you lie with your head on another man's pillow
When you have proved your love so late to me?

-which should probably be followed, as in Mr. Hammond's verse 3:

Now for your sweet sake I'll wear a mournful willow
Now and forever I'll wear it [? for thee.]

The question arises- was a willow-wreathed hat the strange attire, and was the All Round My Hat verse -or something like it- the opening verse of the lover's song in certain versions of The Nobleman's Wedding?

Here, Mr. Freeman goes into detail which need not concern us, and resumes:

As to the comic song of the 'thirties... the All Round My Hat verse is merely the chorus of a song sung in the character of a costermonger whose love is far, far away -for the sufficient reason that a cruel judge had sentenced her to seven years' transportation for thieving- the forlorn coster interpolating his ditty at the end of every two lines by crying his vegetable wares.  So the solitary verse is apparantly the sole relic of an earlier song, seized upon, together with its engaging tune, to provide sport in the music-halls and be whistled by every errand-boy, for it became one of the most popular of street-songs a hundred years ago."

At this point, Mr. Freeman ends by discussing other tune variants, including an untitled tune, no. 119 in Petrie, which is another close relative.  The above is of course largely speculation, though based on what appears to be very reasonable evidence.  It remains to say that the borrowing of tune and verse may perfectly well have been in the opposite direction (more recent authorities might seem to take that view), but here, at least, is food for thought.

See also this previous discussion:  The Nobleman's Wedding