The tune is that of another version, not the one that was used for 'Birmingham Sunday'. 'I Once Loved a Lass' (a Scottish form of the song) is given the same tune in the DT as 'Birmingham Sunday'; unfortunately no source is credited, so don't rely on it being accurate or even the right one.

Here's a provisional midi, made from memory (and so likely to contain mistakes, particularly in the phrasing) of Shirley's recording, which I only have on vinyl at the moment. It may do to start with, though. Corrections would be welcome.

http://folk-network.com/audio/whitsun.mid