According to Hugill, in the British Navy a fiddle was used instead of singing to accompany work, (shanties were generally forbidden) and the fiddler was an official member of the ship's company. In the merchant marine I think shanties were usually unaccompanied, if only because the shantyman was often participating, however lightly, in the work - Hugill expressly says that when pumping the shantyman would take one of the handles. Very occasionally a fiddle might be used to accompany.

Worrall's article says that instruments were used by sailors mostly to provide music for dancing. He also gives figures for the occurrence of various instruments on board ship, taken from a digital search of written references. These show that the "accordion" (probably usually meaning melodeon) appears 331 times compared with 262 for concertina. However top of the list is fiddle (1315 references).

Franklin's predecessor Capt Parry famously took a barrel-organ on his Arctic expeditions in the 1820s. Both the melodeon and concertina are traditional instruments among the Inuit, which were introduced by sailors from whaling ships (and I recall hearing a recording in Ottawa museum of an Inuit song which sounded remarkably like "What shall we do with the drunken sailor?")