In the notes to the 1996 CD by John Fleagle, World's Bliss, Medieval Songs of Love & Death (Archetype Records 60103) he does a beautiful version entitled "The Hern." (The album is a sort of early/traditional crossover thing. Really neat.) He says adapted it from the Middle English Richard Hill's Commonplace Book (c. 1500). Presumably he updated the language a little (though he didn't for other songs on the CD). And it looks like he added a couple verses. Fleagle uses a Breton melody called "Ar falc'hon."
Fleagle says, "'The Hern' is a carol [a dancing song] from Richard Hill's Commonplace Book (c. 1500) which survives in its sung form as 'Down in yon forest.'"
(If you're looking in Leach's book, it's "Over Yonder's a Park" or "Corpus Christi.") The Oxford Book of Carols calls it "Down in yon forest," and in the notes has the same Richard Hill MS text as Leach, which is abbreviated compared to Fleagle's.
In Fleagle's version, the story opens with 3 verses (rather than one) of the hern (heron) flying around before coming upon the hall, and the second refrain line is "The falcon hath borne my mate away" which is a modernization of the Middle English original "The faucon hath born my make away." (The heron's mate has been taken by a falcon -- which I suppose could even have belonged to the bleeding knight...) Here's his text:

The hern flew east, the hern flew west,
Lully lullay, lully lullay,
She bare her o'er the fair forest
The falcon hath born my mate away

She bare her o'er the meadows gree,
All to espy what might be seen.

Oh, then she saw an orchard faire,
Where grow'th the apple and the pear.

And in that orchard stands a hall
Was clad all o'er with purple and pall.

And in that hall there stands a bower
was covered o'er with periwink flower.

And in that bower there stands a bed,
With silken sheets of gold so red.

And in that bed there lieth a knight,
Whose wounds do bleed both day and night.

Under that bed there runs a flood,
One half runs water, the other runs blood.

By the bed side there stands a stone,
A leal maiden was set thereon.

With silver needle and silken thread,
She stems the woulds where they do bleed.

~ Becky in Tucson

Judy - my mother finally found the CD's I left with her last summer to send to me -- including yours! I'm looking forward to getting my hands on them again.