Mueller seems to be a common name with Stockel showing 3 pages of gunmakers by that name, it shows Bernhard Mueller of Zurich 1867-1955 being active.
m-4
I thought Mueller was probably Swiss. The area of Ferlach, Brescia, and the Swiss gun-making areas are not that far apart. They make up a sort of cluster. I did some research on a Ferlach gunmaker who is a distant relative and I saw the migrations of the others of the so-named Ferlach family over the years primarily into Switzerland and then Germany. Germany is not that far away either. Muller, etc, is a common name (Smith) so there is no doubt there could be a family of gun-makers.
My feeling is this Mueller in question was established in London but returned to Switzerland when the Great War was looming. Wouldn't you?
m-4: I am not familiar with the work by Stockel- can you give me more information?