The Kingdom of Great Britain resulted from the political union of the kingdoms of England and Scotland with the Acts of Union 1707 on 1 May 1707 under Queen Anne. In 1801, under a new Act of Union, this kingdom merged with the Kingdom of Ireland to create the United Kingdom of Great Britain and Ireland. After the Irish War of Independence most of Ireland seceded from the Union, which then became the United Kingdom of Great Britain and Northern Ireland.<<<< Wiki
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Lord Cochrane was later than either of these first two formulations, & therefore arguably British, and could be said to have fought for Great Britain, or even the United Kingdom in its original form.

Possibly the lack of song is due to his having subsequently fallen out of favour, dismissed the service and falsely imprisoned for fraud ~~ though later receiving a royal pardon. In the interim between release and pardon he fought for S American navies, receiving a Brazilian marquessate which, tho not strictly hereditary, appears still used by the Earls of Dundonald {the Wiki entry under Dundonald most informative}. I can't see, though, on second thoughts, that these misfortunes should have inhibited any ballad-making contemporaneous with his victories, and agree that the lack of songs about him is surprising.

~Michael~