Its impossible to be terribly precise without seeing the pistol but...
Lack of proof marks means nothing. Most K export guns (but not all of them) were proofed. Proof simply was not required for any gun being sent out of the UK but was commonly expected by the customer.
"T. Ketland & Co" refers to the partnership between Thomas Ketland Sr. & William Walker although Walker's name does not usually appear until the very end of the firm. The company was in business under that name from at least 1785 to 1821 (although TK Sr. and Walker both died in 1816). The Ketland in America, Thomas Jr., was never a gunmaker in any sense. He was a merchant prince. He was not the agent of his father's company but a partner in two other firms, that of William & Alexnder Walker (his father's partner in the gunmaking firm) and Ketland Cottrell & Co. After about 1804, TK guns were sold through other hardware dealers as TK Jr. nearly went bankrupt and stuck his father and Walker for a considerable amount of money. I seriously doubt that any member of the family had any "hands on" experience with gun making after about 1780. However, TK Jr. returned to England in 1815, after the War of 1812, and it looks as if he took some part in the management of the company with John Adams (It was Ketland Walker & Adams near the end, then just Ketland & Adams). Adams was TK Sr's brother-in-law and TK Jr's uncle.
However, the export of British guns to America in sizable quantity does not start until 1792 when the Privy Council started giving licenses for the export of "military stores." Prior to this, while some bits certainly came in, they were likely on a very small scale and the scant surviving records on the British side do not tell us what they were or where they were being shipped. Early on, the K&W firm did a considerable amount of business shipping guns to Ireland (which attracted official curiosity at one point)... so we can't presume that everything they exported came to America.
One of the main export products was gun locks. Tens of thousands were imported of all grades. The cheapest had no internal or external bridles but this is a reflection of price rather than when they were made. Locks of this grade were still being exported in large quantities after the War of 1812.
The wood is inconsequential as well. One of the main exports to England at the time was hard wood, walnut especially but I even have proof of TK Jr. exporting maple stock material to England. TK Jr owned hundreds of acres of hardwood forest in Pennsylvania.