It's in D-sharp minor.
For those of you who're playing along at home, that's six sharps. The melody is entirely pentatonic, which means that on a piano, it comes out entirely on the black keys. Hence the muted, wintry feel that pervades the song. D-sharp minor (or, its relative major, F-sharp, which is the same thing) is also not a tuning one just stumbles across on a guitar, either. This was a very deliberate choice on the Ramblers' part.
(Digression: I've always been baffled by the way in which different keys have different feels, even on a keyboard instrument tuned with perfect equal temperament. By any mathematical reasoning, they shouldn't, especially for those of us not blessed with perfect pitch. And yet they do. D major is bright and energetic; C minor is heavy and oppressive. And just about everyone trained in piano I know can easily tell a tune mostly in white notes from a tune mostly in black notes -- even when they're off by as much as it's possible to be off by in picking out which particular black note key we're talking about.)
Anyway, it's a good song, and I don't feel quite so bad at my initial confusion trying to transcribe it. When I sat down, I just assumed it would be in a nice easy key, and grew increasingly frustrated at my inability to play along. True, my bafflement speaks ill of my long-term memory and reveals just how much of my musical education I've managed to slough off ovet the years, but I'm not quite as musically senile as I'd feared.