Steeleye, used two separate songs. The first is based on the Barrow boy-Penny Gaff-Music Hall rewrite of the Nobleman's Wedding. Hence the 'All around my Hat reference and the tune. The second is based upon the verses for 'Farwell he/she, which may be found in numerous collections.
If you wish to hear the original listen to Joe Holmes or Eddie Butcher for the first and Cyril Tawney and many others for the second.
Chase up the Barrow Boy version in Ballads online, and you will find it waiting complete with street cries 'Get your cauliflowers here!'
Seems to have been a happy and lucrative musical marriage, so no criticism intended.