The firm of Rotherham & Sons, based in Coventry (Warwickshire, U.K.), originated in 1822 when John Rotheram (note spelling) joined the 75-year-old firm of Vale & Co. as partner. By the 1880s Rotherhams were operating on a very large scale by British standards, with over 500 employees active in the production of both movements and cases. The firm maintained some kind of existence until as recently as 1973, but most of their surviving products seem to date from between about 1860 and 1914; later items, and some earlier ones too, are likely to be bought-in Swiss movements.
To arrive at a date for this watch we would need to see some images, especially of the interior of the case (which may well be silver and should then carry 'hallmarks' which would give an exact year) and of the backplate of the movement*. I cannot see that any research has yet been done on Rotherham serial numbers, but even without a hallmark it is usually possible to date an English movement to within twenty years by minor points of style; for example, if it has a compensated balance (with screws around the perimeter) it is probably after 1890, since British makers were slow to bring this component into general use.
I am sorry I did not notice this query until today.
*See the top of this page for instructions on posting images. If no digital camera is available, one can usually obtain a serviceable image by simply placing the watch on an ordinary flatbed scanner.
What I don't know is whether this dating will get you the age of the works as opposed to just that of the case. Were this a US watch, we would not use the case to date it. It could be that Rotherhams made both.