The web site Band of the Irish Guards has a web site:
They have a CD out which has the tune that you are looking for, "St.
Patrick's Day in the Morning" which evidently has been around since at
least the 17th century.
The Band Of The Irish Guards
REGIMENTAL MARCHES (1 .55)
Let Erin Remember
St Patrick's Day
The Regimentol Slow March of the Irish Guards Let Erin
Remember first appeared in Bunting's'Ancient Music of
Ireland' under the title 'The Little Bold Fox'. The lyrics 'Let
Erin Remember' were added by Thomas Moore.
The Quick March St. Patrick's Day probably first appeared in
print in 1749, although it is thought to have been known as
early as 1615.
THE BAND OF THE IRISH GUARDS
Director of Music: Major M.J. Henderson psm
Bandmaster: WOI B. Wassell ARCM psm
(Irish but not a Guardsman)