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.
Thank you for providing very-precise measurement of the ball's diameter, which (as I've mentioned) is crucially necessary to get you specific identification of the projectile. Your excavated lead ball's .4895-inch diameter means it was for a Colonial-Era-through-early-1800s .50-caliber musket or rifle... for example, a .50 Hawken Rifle or a .50 Kentucky Rifle.
I should mention, I included the word "excavated" in the above info because being excavated affects your lead ball's diameter. Excavated lead usually (but not absolutely always) has about .005-to-.01-inch of Lead Oxide "patina" on it, which adds to your lead ball's original diameter. On the day it was lost, its diameter was approximately .48-inch... which is correct for a Colonial-to-early-1800s .50-caliber firearm.