LD1 & Toby:
Alas, Toby may be correct here. I just checked my 12-bore Mortimer, No. 720X, made in 1888, and see only one spot where a serial number appears -- the guard tang. The barrels carry a 5-digit number and there are a few other numbers here and there, but NOT the serial number. My internal pics do not show one either.

The gun may well have been completely made at Scott's... including the Mortimer name engraved on the sideplates. As Toby says, enjoy it for what it is!

Best, Kensal