No, I too know of no songs about Cochrane. Given the British fondness for flawed heroes and his thoroughly chequered career this is a bit surprising.

He was allegedly an inspiration behind both Horatio Hornblower and Capt. Jack Aubrey, so not ignored by 20th century authors.

I had never heard of the man until I read Sharpe's Devil (by Bernard Cornwell) and can only assume that Victorian sensibilities managed to largely airbrush him out of popular history. If you enjoy a bit of historical fiction the Cornwell book is worth a look. The notes fill in a bit of info on Cochrane's career, and indicate that there is a good biography "Cochrane" by Donald Thomas (Pub. Andre Deutsch, 1978) but I haven't seen a copy yet.