DtG - certainly the merchant fleet was used to transport British troops. The Royal Navy only had a few transports itself (usually converted from 'over-age' or redundant warships), and so hired merchant vessels. For example, there were well over 300 transports hired to convey Sir John Moore's army from Corunna, and ever more for the Walcheren Expedition.
I'm not sure that would explain the song, though. I'm of the blister school myself, though I'm sure that a regularly-employed seaman's hands would be hardened enough to avoid blisters. Newcomers (always a butt of jokes anywhere, but even more so at sea) could suffer agonies before their hands grew the necessary callouses (they were encouraged to soak their hands in salt water and to piss on them!)