According to Wikipedia, London's Drury Lane is named after the Suffolk barrister Sir Robert Drury, who built a mansion called Drury House there around 1500. The property passed out of the family some 120 years later, becoming the London house of the Earl of Craven. He later made it into a public house under the sign of his reputed mistress, the Queen of Bohemia. The gardens and courtyards of the house were subsequently built over with rows of smallhouses. The remains of the house itself, which had been progressively demolished, were finally cleared in 1809. By this time Drury Lane had become one of the worst slums in London, dominated by prostitution and gin palaces.
'Drury Lane' has long been a metonym for the Theatre Royal, which has stood there since 1663 (in several incarnations). The lane's other claim to fame is that J. Sainsbury opened his first shop there in 1869.
Some versions of the nursery rhyme have the Muffin Man living on Mulberry Lane; I've even seen references (in American sources) to Strawberry Lane and Blueberry Lane. But there is little doubt that in the original version he lives on Drury Lane.
Frank Zappa wrote and performed a song entitled Muffin Man, which was clearly inspired to some extent by the traditional nursery rhyme, but otherwise has very little in common with it.
© Haydn Thompson 2021