Ta Mick. And Joe.

Sean,
Yes, both are possibilities then.

I now have a copy of Yorkston's sheet music in front of me and here are all the vital details which you may find useful.

Cover page
Cockles and Mussels, or "Molly Malone" Comic Song BY
JAMES YORKSTON
ENT. STA. HALL.    PRICE 3/-
LONDON
FRANCIS BROs. & DAY, (Blenheim House) OXFORD St. W.
Publishers of
SMALLWOOD'S PIANOFORTE TUTOR, THE EASIEST TO TEACH AND TO LEARN FROM.

Inside cover has 8 snapshot scores of waltzes/polkas etc.,

Page 1
COCKLES AND MUSSELS
Written and Composed by James Yorkston. Arranged by Edmund Forman

(Bottom of page) Reprinted from No.35, Musical Treasury, by permission of Messrs. Kohler & Son, 11, North Bridge, Edinbro.

F&D. 1713

The rest is usual 3 verses and chorus.
NB. pages 3 & 5 (chorus) are identical.

Back page has 8 more polkas/waltzes etc snapshots.

From this I would guess that the Musical Treasury version might prove useful with dating. There is no date given on this sheet and not even an embossed stamp which sometimes contains a date.