Smellie, a nice find! The company has some history. Jones was first recorded in 1826 at 27 Stanhope Street in Liverpool, Robert Jones may have taken over the business of John Jones. The firm moved in 1830 to 32 Stanhope Street, and in 1834 Edward Jones was recorded at that address as well.
In 1846 it appears that Robert moved to 9 Waterloo Road with additional premises at 53 Oldhall Street, but there may have been a son of the same name and this may have been when he took over the business.
In 1855 the firm moved to 32 Great Howard Street.
In 1872 the firm patented a push-forward underlever (No. 2121).
In 1885 together with W Taylor, Robert Jones patented a "comb" top lever (No. 7715).
At some time the firm moved to 42 Manchester Street (N.B. YOUR GUN) where it appears to have remained until about 1908. They appear to have opened premises at 60 Dale Street in about 1902 but also may have occupied 6 Chapel Street for a short time.
Probably during the First World War, the firm became Hooton & Jones, they were still located at 60 Dale Street.
Reportedly, the firm (Robert Jones? / Hooton & Jones? / James H Hooton?) was bought by W C Carswell in 1920, but in 1924 "it" (James H Hooton formerly Hooton & Jones) was bought by W Richards (Liverpool) Ltd.
There were (at least) three generations named "Robert", and all were gunmakers. R Thomas is a descendant of Robert (2) who was his great great grandfather (born in 1833, emigrated to the USA in 1885 and died in New York in 1905, profession on death certificate given as "Jeweller").
So, all told, your Robert Jones will date from no later than 1908, and probably after 1902, although possibly earlier. Most of the above info comes from the Internet Gun Club.
There is paint on the gun? Where? Who would paint an English SXS? Please be very careful in cleaning it up and do not cause more damage.
I would not shoot it until a qualified double-gun gunsmith has looked at it, and even then only with black powder loads or hand loaded low pressure loads.
Are you sure that the chambers are 2 3/4 inch? Remember, an unfired 2 3/4" shell will drop right into a 2 1/2" chamber.Is it a hammer or hammerless gun? Sidelock or boxlock? How much engraving coverage? Can you see the pattern in the Damascus barrels? Is there any choke? Sorry if I am being nosey.
Proof marks on the barrel flats and water table may tell you more of the story. Many English guns were re-proofed along the way, if you can post pics I can help interpret the marks. I hope some of this helps.
Sharptail