There isn't actually a key change, or a switch from major to minor; the melody is in the Dorian mode.  Steeleye Span's harmonisation is perhaps the source of the confusion.  To be fair to them, they simply arranged -quite faithfully, in this case- a traditional version of the song (I would describe their recording as an arrangement rather than a "version" in its own right) which Cecil Sharp noted from the singing of William King, a farmer, at Castle of Comfort, Mendip, Somerset, in 1904.  If anyone has a complaint about the speed at which they sang it, they should take it up with Mr. King; as a genuine traditional singer belonging to a living tradition, who was used to singing the song in its traditional context, I should imagine that he knew what he was doing.  Other versions of the song are known, some of which are taken at a faster pace; they tend not to have such attractive melodies, though.  There's nothing essentially wrong with accompanying such a song; Sharp published an arrangement for piano, and Dr. Maud Karpeles, when publishing the melody, provided suggested chord names.  The important thing, of course, is to ensure that the accompaniment does not detract from the song, or force it into a shape or rhythm which doesn't suit it.  If in doubt, sing unaccompanied, which is how these songs were designed to be sung.

The chords Dr. Karpeles suggested are:

It's a rose (Em) bud in June (Bm) and vio (C) lets in (D) full bloom (E)
And the small (Em) birds sing (G) ing love songs (A) on each spray (E)
We'll pipe (G) and we'll sing (Bm) love, we'll dance (C) in (D) a (Em) ring, (G) love (A)
When (C) each lad (Em) takes (D with an f sharp) his (A) lass (G) all (A) on (C) the green (D) grass (E)
And (C) it's all (G) (C) (G) to (Em) plough (Bm) where the fat (C) ox (D) en (Em) graze (A) low (G)
And (D with an f sharp) the lads (Em) and the (Em) lass (Bm) es to sheep (C) shear (D) ing (G) go. (A) (Em)

Obviously, many of these can be omitted; they were only intended as possibilities.

Malcolm