Introduction: A compendium of American gunsmiths lists William Burnston as a Bedford County, Pennsylvania gunsmith. So far, I haven't found any additional information on William Burnston, and I do not know if he worked within the current bounds of Bedford County.
1840: I did not find an entry for William Burnston in a transcript of the 1840 census of Bedford County, Pennsylvania.
1846: Sellers' 2008 book "American Gunsmiths" lists William Burnston as a gunsmith in Bedford County, Pennsylvania based on an 1846 tax list. Sellers indicates his source for the information is "Dean". Based on Sellers' bibliography, I think "Dean" is a reference to the unpublished material of gun collector Herman C. Dean.