The whole point of the Berkeley Square song is that there wasn't really a nightingale - it might as well have said, "A Unicorn walked in Berkeley Square".
Guest above: I'm ready to accept that it may be just a poetic device, but I wouldn't have thought it would appeal as such to people who were unfamiliar with nightingales.
However, further Googling suggests that they were once (this must be pre-20th century) found as far west as Cornwall, so maybe they were known there at the time the song was created, or in literary/folklore sources dating further back still.
So I'm no longer as puzzled as I was, although still interested if anyone has anything further to add.