When I was in school many years ago, an ex-Dane (taught manual training in the schools) placed a recipe for english sparrows in a pie in the local paper, based on european recipes. At the time there were too many of them around. Look in very old cookbooks and there are recipes for larks and other small birds (the French still set up nets to catch them in rural areas).
This may have nothing to do with the "nursery" rhyme, but Catspaw's BS was correct about blackbird pies.