I have heard an altogether different origin for this nursery ryme. For it to make any sense one must first be aquainted with the following
The blackbird is the animal associated with the IRA (Irish Republican Army)
IRA members routinely fill their pockets with rye seed when they go on a mission. If shot and/or killed in the ensuing action, the rye seed falls to the ground and “Patches of rye grass” are seen the following season (commemorating the loss of the member)
With these two salient facts in mind, (and considering how nursery rymes were often used as a means of making cloaked political statements) It is not a far leap of extrapolation to find a second meaning behind the words of this nursery ryme!
I did the research on that one some time ago, and I don’t recollect why “naughty boys” became “birds.” I personally wouldn’t be happy eating a pie that had a bunch of cooped-up live birds in it, what with feathers and bird shit and all, but tastes vary, I suppose.