After cauterization yesterday, how likely is it that my nose will bleed on-board a plane flight tomorrow?

Use saline gel: After cautery you are likely to have the equivalent of a scab on the inside of your nose where the heat was applied. Healing of a cautery site occurs best in a moist environment as moisture prevents premature separation of the scab (bleeding). Airplanes have low humidity (moisture) and can dry the nose out. Avoid Aspirin and use a saline gel product such as ayr. It is easy to use while traveling.