Though interesting and important, the broadside collection at the Bodleian is not the fount of all knowledge, and you will have to look further afield if you want to get anywhere with this kind of exercise.  Beware of becoming too carried away by any one particular resource!

It has been considered that All Around My Hat was a derivative of The Nobleman's Wedding, from which it may indeed have borrowed a verse and a form of its melody; alternatively, the borrowing may have been in the opposite direction.  There are relevant references in numbers 31 and 34 of the Journal of the Folk Song Society (1927 and 1930) which I'll look up tomorrow if I have time.

Peter Kennedy and others, however, would consider it to be a quite separate song.

The set given in Sabine Baring Gould and H. Fleetwood Sheppard's A Garland of Country Song (1895) is from Fleetwood Sheppard himself; he noted:

"This song is given from my own recollection of it sixty years ago, when it was more than popular; it was the delight -the ecstasy of the London street-boy.  I doubt if it ever penetrated far into the country, its vulgarity was too essentially metropolitan.  The hero was a costermonger, his fair love was a pick-pocket, transported for theft. I have modified two or three of the original verses.  There was no real humour in them, and the London dialect of that day is a thing of the past.  It was the charming air which gave popularity to the song.  Chappell supplies the original form, and says it is a Somersetshire tune.  No doubt he is right, but he has no name for it other than The original of All Around My Hat.  I think there is no doubt it was a dance-tune, but the first strain is omitted from the song."

The tune given by Fleetwood Sheppard is the familiar one.

In his notes to the set given in Folk Songs of Britain and Ireland (recorded from Harry Westaway of Belstone in Devonshire, 1951, again a variant of the familiar tune), Peter Kennedy refers to some other versions in print, and again mentions the tune given by Chappell.  Fleetwood Sheppard's reference was to Chappell's Collection of National Airs (1838-40), which I haven't seen; Kennedy however states that it re-appeared in Chappell's Ballad Literature and Popular Music of the Olden Time (1859), as The Budgeon is a Delicate Trade (1725).  Chappell also mentions the tune as having also been used for The Miller of Dee and a number of traditional harvest home songs; the reference to All Around My Hat is not, however, repeated, nor is the reference to Somerset.

The DT file  THE MILLER OF DEE (Jolly Miller),  taken from Chappell, also points out that the tune was later used for Rolling Down to Old Maui (I suspect I've also heard it used for Edmund in the Lowlands).  This is a rather different, minor melody from that given by Sheppard and Kennedy; Chappell however states that the harvest-supper tune was in the major, and particularly cites the set printed in the Rev. John Broadwood and G.A. Dusart's Old English Songs, which may perhaps be closer to the Hat melody.

In an earlier thread,  The Nobleman's Wedding,  Bruce Olson postulated a link between The Willow Green and The Nobleman's Wedding.  Whether there is any direct link between it and All Around My Hat is another matter, of course.  People have been wearing green willows, and doing things for a year and a day, for rather a long time, and what may at first sight appear to indicate a connection may just as easily be no more than a coincidence, particularly where the expression seems to be a commonplace of long standing.