Mary...'s version looks about right. There's nothing in The Bodleian collection and (oddly) there's nothing in Levy, but The British Library catalogue has two entries for 19th Century versions.

"Cockles & Mussels" Song [begins: "Of all merry blades"] Geoghegan, Joseph B 1876.

"Cockles and Mussels" Comic Song. (Written and composed by J. Yorkston, arranged by E. Forman.) Yorkston. James 1884.

Style is much more 19th Century also.

Cheers!
Ian