You don't HAVE to use oil to quench an oil-hardening steel. Steels which are oil hardening simply have alloying elements which allow the steel to reach full hardness with a slower, less intense quench. Think of it as a minimum necessary to harden the steel - so you could quench an oil hardening steel in oil or water, or an air hardening steel in air, oil, or water.
Most knifemakers here will use oil, since oil is a superior quenchant - you get less distortion and less chance of cracking. You can look at the curves showing cooling rates of various quenchants at different temperatures. An oil quenchant tends to cool much more slowly than water during the martensite formation phase - between about 400 degrees and room temperature.
(Martensite is a hardened form of steel - when you etch the khukuri and it shows a temper line, this is a boundary between the hardened martensite and a different kind of microstructure, known as pearlite.)
When martensite forms, the steel expands, this is what causes the distortion or cracking. The quicker the formation, the greater the chance of distortion or cracking - so water is more risky. You can also interrupt the quench - stop it at 400 degrees and let the blade cool in air. This is called marquenching, and minimizes distortion. Maybe the kamis do this, but I doubt it.
(Water is considered an inferior quenchant for other reasons too - the boiling point is so low that a vapor jacket forms and stays around the thing being quenched, which causes uneven cooling and lowers the cooling rate. This is not so much a problem with oil, which cools much more evenly)
It's just possible that the area that the Kamis actually quench is so small in relation to the huge, thick blade that the distortion effects are pretty much insignificant.
On a related, interesting note, some current research has shown that in fact, EXTREMELY rapid cooling throughout the martensite formation phase may also reduce cracking, and have other beneficial effects, such as increased hardness, wear resistance, impact resistance, etc. This is called "Intensive Quenching". (take a look at
www.intensivequench.com/intensiquench.html).
It was developed to allow a low cost alternative to oil quenches, and uses highly agitated water or brine. The technology was first used in Russia, and had to be low-tech and foolproof.
It would be very interesting if the Kamis could implement a type of intensive quench. Normally it doesn't work so well on objects of thin cross section, like knives, but some of those khukuris are so thick that an intensive quench might just work.
Back on topic, I've wondered about the tempering process myself. I assume that they just temper using the edge of the forge fire. They probably watch the oxide colors forming to gauge the temperature. It's not the best way to do the tempering process, since it could lead to inconsistent results or mistakes. My friend had a Sirupati which appears to never have been tempered. It chips out even on soft material. But, with care, this method ought to work well enough.