What is the first aid when choking on one's saliva?

Most healthy people can swallow their own saliva, so not being able to do so is indicative of a serious neurological insult, that should be taken into consideration.

That said, first aid for choking on one's own saliva is suctioning the saliva out. If you don't have suction with you, and there's no other injury that precludes this, roll them into Rescue Position to the head faces to the side and the mouth faces down. Swab excess saliva from the throat, being careful not to trigger the gag reflex.