The title of this tune does not refer to the jacket worn while peeling potatoes, carrots, or cucumbers. Rather, it is an eponym (not an aptonym!), since part of a person’s name is used to refer to a thing, specifically an occupation; but it is also a synecdoche since one man (a part) is used to refer to a group (the whole). Last, a jacket is . . . we, you know what a jacket is.
In 1829, Robert Peel, who was Home Secretary in the British Government Cabinet of the day, set up the London Metropolitan Police Force, the first properly organized police force in the world. Henceforth, policemen were often called “bobbies”, after Robert Peel’s name, and also “peelers”, but possibly less commonly.
The tune is sometimes known as “The Flannel Jacket,” but that is not as fun.  A very nice version of this tune can be found on the album Up Close (1984), by Kevin Burke (fiddle) with Joe Burke (accordion).