There's a variant on this song referred to as The Somerset Light Infantry on an album of regimental music. Can be heard here:
Much faster tempo than the version used in the movie, but definitely the same tune. Can be purchased on Amazon or iTunes.
"The melancholy truth was that his glorious golden head had nothing in it."

Yes, the Prince Albert March seems to be correct.
And, unfortunately, you are correct that there seem to be no recorded versions of this jaunty tune anywhere.

I believe he's referring to the song played during the march to the Alma immedeately after landing. The same song was played at the beggining of the movie as the regiment was being paraded in England. The song is "The Prince Albert March" and near as I can tell it is a period piece that was actually used as the 11th Hussar's Regimental march (they were Prince Albert's Own, remember?) but it seems to be very obscure otherwise. The fact that the regiement no longer exists probably didn't help matters. I've only managed to find sheet music of it online, not even a midi. The sheet music I found was arranged by one Philip Ernst.

Im no expert on British Marches, but it is likely to be one of these.
The Infantry march into battle at Alma whistling, The Girl I left Behind Me, a traditional Irish air.
http://military-bands.co.uk/quick.htm
THE QUEEN'S ROYAL LANCERS - Stable Jacket; THE QUEEN'S ROYAL HUSSARS - The Queen's Royal Hussars; THE KING'S ROYAL HUSSARS - The King's Royal Hussars; THE PRINCESS OF WALES'S ROYAL REGIMENT - The Farmer's Boy/The Soldiers of the Queen; THE ROYAL GLOUCESTERSHIRE, BERKSHIRE AND WILTSHIRE REGIMENT - The Sphinx and Dragon; THE HIGHLANDERS - The Queen's Own Highlanders & Cock of the North; THE BRIGADE OF GURKHAS - Yo Nepali; THE ROYAL LOGISTIC CORPS - On Parade; THE ADJUTANT GENERAL'S CORPS - Pride of Lions; CORPS OF ARMY MUSIC/RMSM - Blow Away the Morning Dew; THE QUEEN'S DIVISION - The Queensman; THE KING'S DIVISION - Bond of Friendship; THE PRINCE OF WALES'S DIVISION - God Bless the Prince of Wales; THE LIGHT DIVISION - The Light Division;