Claude disagrees a bit:
The Maker: "Mash, 10 W. Wigmore Street, London"
This almost certainly refers to George Mash, a London gunmaker who traded from Wigmore Street in the latter half of the 19th century. Wigmore Street was a respectable but not quite "best" London address — not Bond Street or St. James's, but a legitimate West End trade location catering to a solid middle-to-upper-middle-class clientele.
George Mash is documented in London trade directories of the 1860s–1880s period. He was a retail gunmaker and agent rather than a manufacturing "maker" in the strictest sense — meaning he likely had guns built to his order by Birmingham or London trade manufacturers and retailed them under his own name, which was absolutely standard practice. The "10 Wigmore Street" address helps narrow the date; London gunmakers moved around, and pinning an address to directory records can give you a manufacturing window within a decade or so.
The abbreviated "Mash" on the rib (rather than a full name) is consistent with rib engraving conventions where space was limited on a straight, narrow top rib.
The Gun Itself
29" barrels on a hammer gun of this era are a very typical and practical length — neither the longer wildfowling barrels nor a shortened game gun. This is a standard all-round game gun length.
Damascus barrels described as "beautiful" suggests a quality iron/steel weld pattern — likely a multi-iron twist or fine Damascus rather than common twist. On a London-retailed gun of any pretension, the barrel quality would have been a selling point, and good Damascus from this era is genuinely attractive, often with a fine figured pattern when properly browned or re-browned.
"Great dimensions" — I'd love to know the specifics. Length of pull, drop at comb and heel, cast? These details would tell you a lot about whether this was a bespoke or semi-bespoke piece vs. a standard off-the-shelf stocked gun.
The Stock
A well-executed stock extension is not a red flag on a gun of this age — it's actually rather common and speaks to the gun having been used and valued enough to be properly fitted to a subsequent owner. A clumsily done extension devalues and weakens; a well-done one, properly fitted and finished, is essentially neutral. The inlaid steel buttplate is a nice period detail and more durable than a horn plate.
Dating
Without proof marks or action serial numbers, a Wigmore Street Mash hammer gun with Damascus barrels most likely dates to circa 1865–1885, with the probability highest in the 1870s. By the late 1880s, hammerless actions were becoming fashionable among London retailers, and hammer guns were increasingly the economy or conservative-buyer option.
What Would Help Narrow Things Down
Proof marks on the flats — London or Birmingham, and the specific mark (view + definitive, and any reproof marks)
Serial number on the action bar or other parts
Lock style — bar-action or back-action? Rebounding locks?
Choke or cylinder boring stamped on the flats
Action maker's name anywhere — sometimes the trade maker is stamped on the water table
Do you have any of those details? Particularly the proof marks — those are often the most reliable dating tool on a gun like this.