Brass 1-piece flatbutton with an indented-lettering backmark including the word "London" dates it from about 1810 into the late-1820s/early-1830s. The word "London" of course means it is British-made. Importation of these simple 1-piece brass flatbuttons from Britain into the United States declined drastically when the US button-making industry finally became capable of mass-producing them, about 1830.
The (presumably) lead ball is perfectly-round, so, given that it is the correct size-range, is almost definitely a musket ball or a pistol ball. Need very precise size-measurement to be able to identify its caliber. It is probably from the same time-period as your British-made brass flatbutton, but of course could be a lot earlier.