Based on the dimensions, your bayonet is likely a Spanish M1857 socket bayonet. It is too short and muzzle ring too small for the Drake Pattern bayonets used with either the Sharps or Mississippi rifles. The Spanish M1857 is the "other" bayonet type known to exhibit the Drake Pattern features. Reilly's book on American socket bayonets indicates that some of the Spanish bayonets are known to have been manufactured in England and Belgium, so the Birmingham mark on yours fits. Spanish M1857 bayonets are scarce, but not as desirable in the USA, as the Drake variants used with American Civil War arms.