The Waterson version appears to derive from the Sharp Mss collected from William Chorley at Bridgwater, Somerset, in 1907. It is indeed accumulative.

The copy given in Reeves, Idiom of the People states

'In this cumulative song, singers started either at the 'first day' and added one line at each repetition, or at the 'twelfth day' and omitted a line at each repetition and then built up again from two to twelve.'

Reeves gives another version from a music book which had been taken down from an old man aged 85 in 1923.

Have you considered it might have been a parody on 'The Twelve days of Christmas'?