Thanks for adding the information about the backmark after I posted. That definitely helps with the date range. My guess would be c. 1820's-30's; they were listed as manufacturers of "gilt & metal buttons" from 1805 through 1833.
Thomas Ledsam & Sons, a British firm with operations in both Birmingham and London, was in business at least as early as 1797. The button presumably dates no later than around 1850, when Ledsam & Sons was bought out by Piggot & Co.
Thomas Ledsam was also listed as a gun maker in 1785, and his son Joseph was listed as a maker of watch chains and toys in 1797.