A lot of the choices in this interesting thread seem to me to be 'grisly' rather than simply 'sad'. I'd second Carthy's 'Bill Norrie' and 'The Banks of Green Willow', though.

No-one has yet mentioned 'The Banks of Red Roses' as sung by Sara Makem (and later by Pete Coe). It's only one song amongst many describing the murder of a young woman by her lover, with unwanted pregnancy implied, but it's powerfully told and set to beautifully plaintive tune. The murder is described in cold-blooded detail, but it's the recurrence after the event of the refrain "Oh my Johnny, lovely Johnny, don't you leave me" that really gets to me. She loves him even as he plans her death, and believes until the last minute that they're just going for a country stroll. Heartbreaking.