I was just hunting, and have discovered that the topic was covered by Malcolm Douglas in a thread on the Deeds of Napoleon in 2005 - and he quite specifically recorded that none of Burstow's recordings had survived. I have a feeling you asked me about that song Tim - not one that I have ever come across. Mind you, a lot of the 400 odd songs that Burstow listed are not "folk" songs as such - a lot are music hall songs. I wonder if the one you're after might have been one such?
The trouble is that Lucy was only after specific songs when she approached him ... modal, and ballads for preference, and announced that only 25% of his repertoire was worth taking down. As Vaughan Williams had worked closely with her at the outset, I wonder if he took the same tack, in which case the song might very well not have made it through the quality sieve?