BSA (lincoln jeffries pat) air rifle from 1905. First batch of BSA made rifles using the Lincoln jeffries design. Should be some writing on the cylinder top. Most likely .177?
As to value, it depends really......
Yes it is a very early rifle which makes it desirable but it is beat up. Someone has added an extra screw the trigger tang which is a shame. Woodwork has some damage and it looks to be missing the wheel for the rear sight.
Not the worst condition one I have seen though. Defiantly work "fixing up". Parts are easy to get from John knibbs (well most parts).
Collectors would tell you NOT to get it reblued, and I would agree, but its up to you.
All I would do is replace missing or broken parts if able and get some 0000 grade wire wool and some gun oil and gently scrub the metal work to get rid of some rust.
Very easy to work on. At one point I must of had about a dozen of these types of air rifles (Which believe me is not a large amount compared to others on here!), ranging from early 1906 to the 1920s dates.
So value I would guess could be anywhere from £100 to £200 in this condition. Others may think differently.
So yeah it is worth getting it serviced. They make cracking rifles even when neglected like yours (My first one was my Grandfathers first air rifle, a CS prefix which had been in the corner of his office for 30 years and even though it was in similar shape to yours, still shot accurately)