In a languagelog blog post, one of the commenters found in Google books a 1903 shorthand book where the pound sign # is mentioned, and it appears that even at the time, it was in the U.S. a well-established symbol for both number (#2) and weight in pounds (5#), depending on whether it appeared before or after the associated number. I expect that this usage developed in the U.S. some time in the 19th century.
It seems to have never appeared in print in the 19th century, presumably because printers used the symbols № and ℔ instead, but I would assume that it was widely used in the U.S. in handwritten documents and signs. (I am fairly sure it still was when I was young, in the 1960s and 1970s.)
The word "octothorp" started in the 1960s with a practical joke among engineers at Bell Labs that got out of hand, after which the perpetrators of the joke were apparently reluctant to tell their bosses that they had inadvertently unleashed a new word upon the world. Many, many years later, after everybody involved had retired, one of the engineers involved wrote up a detailed account of this story: The ASCII Character Octatherp, by Douglas Kerr.
The standard theory of the origin of # is that it is an alteration of ℔. This theory is certainly plausible, because it's quicker to write and looks somewhat similar. However, I don't know if there's any definitive evidence for this theory. Since the use of the # sign for weight in pounds seems to be unknown in England, this alteration probably happened in the U.S. some time in the 19th century, in handwritten documents, and this is the kind of thing that might be very difficult to catch happening in the historical record.