I have a feeling something more sinister is at work here.
When I was a lad, in the 1960s, we learned the song from our parents and grandparents, and pronounced the street name as they did - 'Jewry Lane'. The first written version of the rhyme in the 19th Century mentions Drury Lane, but it is well-known that such rhymes were often written down only after centuries of use, so which is the 'correct' version must be an open question.
'Old Jewry' is a street in the City of London which is the last remnant of a Jewish Ghetto that was demolished in 1290.
'Muffin' is related to the German 'muffen' - 'small cakes'. Though nowadays the word specifically refers to either a small flatbread, commonly split and toasted (known in the US as an 'English Muffin') or a heavy-textured, oversized, over-sweet fairy-cake, it is possible that, to Saxon Englishmen, back in the day, it simply meant 'small cakes'.
Could we be talking about the so-called 'Blood Libel' here? The then-popular belief that Jews were required to periodically kidnap and sacrifice a Christian child? Is the rhyme a warning to children not to accept the offer of cakes from Jewish bakers?