Wm Moore & Grey (gunmaking business under various names from 1820 to 1908) did of course have a Birmingham workshop (Whittal St, Loveday St) at the same time as their London premises, so it would be quite feasible for this gun to have come out of their shop there. I believe most of these guns ended up for export, with higher end guns made in London.

Serial numbers may help to narrow down the dates and from my own research I have the following:

No 374 Earliest number at 78 Edgware Road address, from 1828
Became William Moore & Co in 1836
No 875 Earliest number at 43 Old Bond Street, from 1854
Became Wm Moore & Grey in 1862
No 4737 Earliest number at 165 Piccadilly, from 1896
No 5613 Earliest number at 8 Craven St, from 1902

The proof marks will help narrow down the date for this gun further, since 1862 - 1896 is rather a large span!

Tim