Notes from the Niagara County Cemeteries book (put together decades ago) mark an Old Burying Ground (also known as the Town Burying Grounds) that was once found between 2nd and 3rds streets in Niagara Falls. The book says that the cemetery was removed for the railroad and that bodies and some monuments were taken to Oakwood Cemetery.
Above is a 1856 map that still shows the Old Cemetery even though the rail road is drawn through it. My assumption is that the bodies and headstones were then removed prior to 1856 but that the map drawer still decided to include the cemetery on the map (which I appreciate).
To view the 1856 Map of the Villages of Niagara Falls & Niagara City, New York, click here: https://www.loc.gov/resource/g3804n.la002331.