The earliest form of the rhyme was found by James Orchard Halliwell in his research for his classic "The Nursery Rhymes of England" (3rd ed. 1846).
He found it in the Sloane MS of 1487 (time if Charles I).

Hic, hoc, the carrion crow,
For I have shot something too low:
I have quite missed my mark,
And shot the poor sow to the heart;
Wife, bring treacle in a spoon,
Or else the poor sow's heart will down.

From Halliwell, J. O., 1859, "Popular Rhymes and Nursery Tales, A sequel to 'The Nursery Rhymes ..."
The version printed by Halliwell (four verses) in his "Nursery Rhymes ..."

Lyr. Add: The Carrion Crow
Halliwell, 1846, XCV.

A carrion crow sat on an oak,
Fol de riddle, lol de riddle, hi ding do,
Watching a tailor shape his cloak;
Sing heigh ho, the carrion crow,
Fol de riddle, lol de riddle, hi ding do.

Wife, bring me my old bent bow,
Fol de riddle, lol de riddle, hi ding do,
That I may shoot yon carrion crow;
Sing heigh ho, etc.

The tailor he shot and missed his mark,
Fol de riddle, lol de riddle, hi ding do,
And shot his own sow quite through the heart,
Sing heigh ho, etc.

Wife, bring brandy in a spoon;
Fol de riddle, lol de riddle, hi ding do,
For our old sow is in a swoon,
Sing heigh ho, etc.

(If the sow is shot through the heart, then the brandy must be for the tailor, don't you think?