Fun fact: the name John isn't actually a short version of Jonathan, as one might think. Instead, it's the English version of the Hebrew name Yoḥanan, meaning "G-d has freely given", roughly. Jonathan is also of Hebrew origin, meaning "G-d has gifted", more or less.
The h in John is actually a remains from the ח in the name it came from - a letter that isn't present in Jonathan. Which actually explains why Jon as a nickname for Jonathan shouldn't have an h in it.
The interesting part comes when you realize that the reason both names start so similarly is because they both are theophoric names that start with G-d's name - and I do mean the Name, not the Hebrew word for god. Other names that are similar include Joachim, Joshua (technically Jesus as well), and possibly a couple of other names that are common in English.
This leads to the conclusion that Jo is the most common way to transcribe G-d's name in English. Which is kind of funny, because in the Mishnah, on one occasion when G-d's Name sort of needs to be uttered, the stand in written is "Yose", a common shortening of Joseph during that time. So, kind of like Joe.