I may be missing something (nothing new there!) but this seems to be a Lincoln from the air cylinder onwards, judging from the cylinder markings and knurled tap screw. However, the trigger block and stock appear to be from a BSA Improved Model. I'd roughly date the Lincoln to 1906 (from the knurled tap - roughly, as I say as the option of this type of tap was offered as a retro fit by LJ, so could arguably be found on an upgraded BSA) and the BSA to 1907 (from the SN).

Bearing in mind parts can interchange and the number of old rifles that are parted out on t'bay for example, it would not surprise me if a previous owner made one working gun from two non working guns.

Without the full story, we can only guess from the evidence before us.

John