The name John Buckingham dose not appear in any of my reference books pertaining to the English gun trade. In all probality, it was a gun made by the Birmingham trade and sold under the name of a hardware store[Ironmonger in the U.K.]
Details of the proof marks will help to establish the origin of your gun, can you post same for review?
An outside possibility exists that the name Buckingham relates to a city of that name in Buckinghamshire, England.However the only gunmaker listed in Buckingham is John Meeham, Bridge Street, in 1852


Roy Hebbes