NOT sharp. Strong opinion.

My go-to (or at least first thought) lyric and music for Greensleeves is the version Debbie Reynolds sang in “How the West Was Won.” It definitely needs the sharp.

It's a modal melody and the sharp is to 'modernise' it - just like what was visited upon plain chant in the baroque and classical eras - even Palestrina got caught up in this decadent practice.

An interesting observation, seeing as I hear it as more modal with the raised sixth than without.

... I hear it as more modal with the raised sixth than without.

Indeed, keeping the raised sixth puts the melody in the Dorian mode.

What Mode Is This?

It depends on what theory you subscribe to:

Some would say (and this used to be me) that it is in the First Mode (Dorian), which naturally runs from D-D and does not use the b-flat (except it often does, cf. most Mode I chants in the Graduale); therefore, when transposed to the usual key of E minor, the C should be sharpened.

Others would say (and this is my opinion now) that the b should be supplied with musica ficta, and a flat added (which would be c-natural in the usual transposition), to follow the Fa super La rule, since the opening phrase only touches the b before descending again: so the La of the natural hexachord (C-A) needs to be supplied with the Fa from the soft hexachord (F-d, with the b-flat). This matches with Kyrie Orbis factor.

NB: this is entirely about the possibility of using or not using ficta in the first and second lines; it should, IMO always be sharp in the third and fourth lines (the Refrain in What child is this), since the mode seems to briefly change from the Dorian to Ionian (or transposed Lydian) at that point.

I hear it as thus:

melodic minor for the first half... it then switches to the key of D Major for the refrain, but begins on the IV chord (of D Major) for the first two measures then returns to E Minor for the last two measures of the refrain. You can call it 'modernization' if you want, but I like this approach.

(In the melodic minor, the sixth is natural when descending and sharped when ascending.)

e,g,a,b,c,b,a,f#,d,e,f#,g,e,e,d#,e,f#,d#,b (melodic minor)
e,g,a,b,c,b,a,f#,d,e,f#,g,f#,e,d#,c#,d#,e,e (melodic minor)
d,d,c#,b,a,f#,d,e,f# (D Major, beginning on IV)
g,e,e,d#,e,f#,d#,b (melodic minor)
d,d,c#,b,a,f#,d,e,f# (D Major, beginning on IV)
g,f#,e,d#,c#,d#,e,e (melodic minor)

(It actually reflects a 'mixolydian schizophrenic complex' traveling from E to a lowered seventh (D) in its harmonic progression)

Split the difference! (purple)

My only strong opinion on this is that it should match what is printed in the music. A certain Cathedral would print the music with the sharp and then play it without which was very difficult for those who actually read.