Justin
The name of Smith is well known name of which there were many in the flint and percussion period,both in London and more so in Birmingham the name was also spelt Smithe and Smyth .Most likley your gun was built by London gunmaker James Smith of 19, St Martins Lane 1796 and also 50 Chambers St 1798.
Has for your gun this has been restored with certain things added ,the cock is repro and more assoiciated to a miliatary cock it should have been flat faced for a sporting gun ,the fizzen is also a repro the checkering is also wrong and not the English flat top .
Feltwad