To install a Leather NATO strap, you will need to remove the spring bars from the watch. Otherwise, there is a high risk of damaging the surface of the leather strap if it is "pulled through" the spring bars (as a nylon NATO typically is).
Using your spring bar tool, remove the spring bars from the watch. Separate the two layers of the NATO by pulling the long end through the floating keeper. Next, place the strap within the lugs of the watch, making sure it is oriented and positioned properly. Working from the back side, compress and re-install the spring bars into the lugs to secure the strap in position. Lastly, pass the long end of the strap through the floating keeper again.
This method takes a little longer, but it ensures you won't end up with any damage to the leather from rubbing on the watch case. This method is also preferred for nylon straps being installed on a watch with minimal clearance between the spring bars and watch case.