I might be the closest but I'm not that close, Drury Lane is about 12 miles from my house as the dragon flies, so I took a journey along the place vi Google street view and it's a typical boring London street with mostly offices and blank walls. There are a few shops and cafes, mostly expensive looking places but then again you don't get cheap eating places in London, but I saw no muffin man or his shop.
If you do go to London there are a lot more interesting places to visit than Drury Lane
If there was a muffin shop there and I went in and showed that picture and asked if that man was there I'd get a very rude, and unrepeatable, reply and would probably be thrown out of the shop.