According to WorldCat.org, the British Library has sheet music described thus:

THE CONSCIENTIOUS OBJECTOR'S LAMENT
Words and music by Davy Burnaby and Gitz Rice
London : Herman Darewski Music Publishing Co, [1917]

The catalog gives no further information.

The McGill University Song Book, Vol. 1 (Montreal: The Students' Council of McGill University, 1921), page 193, contains only the lyrics of this one verse:

CONSCIENTIOUS OBJECTOR'S LAMENT
By Gitz Rice

Call out the Army and the navy ('ave a banana)
Call out the rank and the file
Call out the brave old territorials
They'll face the danger with a smile (I don't think)
Where are the boys of the old brigade
They made old England free
Call out my brother my sister and my mother
But for gosh sake don't call me.