Gavin. If it's any help these boxes are simple compared to the majority of contemporary designs.
Make sure you have the right parts. Plunger Layshaft has a plain shaft at the blind bush end, S/A shaft is the same dimensionally, but has a scroll here. They do interchange, providing you use the correct matching pair of layshaft pinion and layshaft third gear cog. The pinion drive dogs have a different profile, square v/s rounded, and the locating circlip position is different.
If changing mainshaft and layshaft pinions they are pressed on with a lot of force. Make sure any oil holes in the gears match corresponding oil holes in the layshaft. Position the sliding gears onto the shaft splines in a similar way so the oil holes can line up with the oilway in the layshaft centre. Kickstart Quadrants must be used with their matching pinions, all previously detailed on the Forum as are the whole gamut of maladies folks have found.
Layshaft endfloat is controlled by a hardened steel washer between the large first gear and the layshaft. This washer has a chamfer on the inner diameter. This faces the clutch, to clear the radius where the layshaft diameter is reduced.
Easiest to assemble shafts, gears and selector arms into the box, leaving off the big first gear, then add the selector shaft rod. Leave the grubscrew 'til the very last.
BSA Service Sheet 209 gives you the basics.
Swarfy.