"Had?" It still does as far as I'm aware. 
But while the chorus is to the tune of "The Sash My Father Wore" - as proddy a song as you could imagine - the verses are to the tune of "Jarama Valley," a heroic song of the anti-Franco Republican forces in the Spanish civil war. Not too many Orangemen there, methinks. 
So, in "Poor Scouser Tommy" we have an ironic synthesis of differing religions, a dialectic for our time if you will. 
But isn't ''Jarama Valley'' a copy of 'Red River Valley', which is an American Civil War tune, therefore mid 19th century?
And the 'tunes' debate could go on for ever. The Bitters ''Grand Old Team'' for example, is actually Celtic FC's oldest remaining tune (after they nicked YNWA that is!); We as a club have songs which swing either way.... PST; 'Bill Shankly of Glenbuck'; ''Soldiers are we'' etc.
Don't think the tunes we sing today reflect in any way the original religious affiliations of the 2 Merseyside clubs (if any ever existed in the same way as in Glasgow or Edinburgh for example).
But there was, in the early days, a tendency to associate Liverpool FC with the Orange tradition, as I'm certain Robbie will/can confirm.
No matter nowadays. If any of them is disaffected, and wishes to move upward to the Higher Level of worship... we'll adopt them all. As long as they leave the baggage of their miserable past behind, honour the traditions of Liverpool and the KFS, condemn green or blue colours to the dustbin.... and learn the various songs without fear or favour.
