Thanks for all of the suggestions, but Mom and Dad identified it as the bridge in the middle of a piece called "Fantasia on Greensleeves," which was adapted from Richard Vaughan Williams' opera "Sir John in Love," by Ralph Greaves.
The piece is two straightforward renditions of "Greensleeves" that bookend the bridge section in which the melody I hummed is repeated with violins (not violence), then with flutes.
EDIT: Have a listen, here (thanks evilnel for the link!).
Here's some additional detail I found through the power of Teh Internets:
The bridge section is based on a traditional English folk song called "Sweet Joan" (which sounds, disappointingly, nothing like "Sweet Jane"):
Lyrics here. Sweet Joan was a sneaky one.
Like to hear it? Here it goes.
And here's a more orchestral version rendered ... um, on what sounds like a Casio.
Truthfully, I'm a little disappointed. I thought the melody suggested something more sweeping and grand, like a symphonic theme or something with a full orchestra exploring all kinds of variations and ... stuff. I dunno, I envisioned Valkyries. Instead, it's three or four measures in a short piece based on an opera. Maybe I can content myself by thinking that Sweet Joan wore a horned helmet and looked like this.