Not jigsaw, it's done on a bandsaw with a narrow blade. The stock is held in a plywood jig that keeps it true and square to the bandsaw blade, and prevents it from rocking while the cut is made. That way you get a good edge with very little dressing to do. I also have jigs to hold the stock in place under a pillar drill while the holes for the riser stems are drilled, and another jig to hold the cut-off cheek piece so that the stems are positioned at the correct centres, and true. If the stems are out of line or not true, the cheek piece won't go up and down smoothly, even if they are exactly the right distance apart.
Finishing, well, no two Air Arms stocks ever seem to be the same colour, and the timber always varies a bit. I sand and wire wool the surfaces, then make up a spirit wood stain of the right colour by blending two standard Colron colours. Err on the side of darker as the cut surfaces are usually in shadow. Finally I rub in some Black Bison wax polish.
Here's another one; shows the riser stems and the bushes.