Cecil Sharp collected music at Kirby Malzeard, but I don't know if that's published. At any rate the 'calling on song' from there is given with music in Broadwood and Fuller-Maitland's 'English County Songs'. The dance was said to be to the tune of "My love she's but a lassie yet". However, Conrad's text above commences with what seems to be what B&F-M call the Prologue, and they gave no music for it (saying it was of a Morris Dance character). They also gave some verses (without music) of another version from nearby Ripon. The prologue is given as:

These are six actors bold,
Never came on stage before;
And they will do their best,
And the best can do no more.

[This looks like the begining of Conrad's above, but it continues:

You've seen them all go round,
Think on 'em what you will;
Music strile up and play
"T'aud lass fra Dallowgill."

And that's all, whereas the first 8 verses of the 'Bishoprick Garland' text seem to be a different 'calling on song'.