The tune given above ("Roud 511; Laws K40; Ballad Index LK40; trad.", not identical to the one to which 'Cruising Round Yarmouth' is normally sung) sounds traditional enough to alert our Shanty Police when it comes to chords.

Certainly we cannot speak of the chords. I did not hear any recording; here is my own suggestion, to be used at own risk:

X:1
T:Dumiama Dingiama Dumiama Day
M:3/4
L:1/4
K:Am
E| "Am" AA "(Em)"G| "Am" A2 G/2A/2| "G" BAG| "C" E2 E|\
"Am" A,/2-B,/2-C/2-D/2 E/2F/2| "C" GEG| "Dm" AGF| "Am" E2 "(G/B)" D|\
"C" E2 E| "G/B" D2 G| "Am" E2 E| "G/B" D2 G|\
"C" E2 E| "G" D-G F| "Am" E-D C| "E7" DCB,|\
"Am" A,3/2 B,/2C/2 D/2| "C" E3/2 F/2E/2 D/2| "E7" E3/2 D/2C/2 B,/2| "Am" A,2 |]

To fit the basically merry lyrics, I took the liberty to use as many major chords as possible, including some (anachronistic) sevenths. Have fun, or ask someone else. To arrange the song for stage performances or commercial recordings, more effort is required.