Prompted by the William Rodgers and sons story I searched the net for John and George Rodgers and, lo and behold, the THE MONTHLY MAGAZINE, BRITISH REGISTER OF LITERATURE, SCIENCES, AND THE BELLES-LETTRES of 1831, vol xi, Jan-Jun, lists, under patents granted:
"To John and George Rodgers, Shef-
field, York, cutlers, and Thomas Fel-
lows, junior, New Cross, Deptford, Kent,
gentleman, for an improved skate.
January 18th ; 2 months."
http://www.archive.org/stream/monthlyma ... d_djvu.txt" onclick="window.open(this.href);return false;
This is confirmed by a second source:
http://books.google.de/books?id=7tQ3AAA ... 22&f=false" onclick="window.open(this.href);return false;
The National Archives list a mansucript of John and George Rodgers as having taken out fire insurance:
John and George Rodgers, Sheffield, Yorkshire, cutlers and hardwaremen
Other property or occupiers: 73 Hatton Garden
So, that even provides an address.
If I get access to the online database of Sheffield cutlers I may learn more, but this seems to me to strongly indicate the mark to have been owned by John and George Rodgers and then taken over by Joseph Rodgers, as you assumed. Of course it still does not explain its appearance on a post-1879 electroplate teapot.