BSA Bottom Bracket Replacement
It seems to me that the thread on either the crank thread or the nut is somewhat damaged.

If this is your daily ride and you are not going to take it for around-the-world-tour, I'd replace the bottom bracket with a cartridge one. It doesn't require any adjusting - great advantage if you are a novice.


Measure the shell width (you can do it without removing the spindle, just flip the bike) and make sure it is 68 (typical BSA). 
Get yourself a crank puller (simple tool for couple of bucks), pull the cranks and measure the spindle length (just measure the tapered parts, discard the threads) and check whether the spindle is symmetrical (i.e. the length from the middle of the bottom bracket to the left and to the right of the spindle is the same).
Check what kind of tool you need to remove the drive side cup. When attempting to remove it, remember that at BSA, the driver side has reverse threading (you unscrew it CCW).
Purchase the stuff you need and put it on the bike. The cartridge bottom bracket has its left and right (driver) side marked.


Good luck!