Japanese didn't have any original letters in ancient times. Kanji was brought into Japan from China in the 3rd or 4th century.
When old Japanese people adopted kanji, they called each letter as Chinese people pronounced it. While, kanji often had the meaning which matched some Japanese words, so they came to read ''山'' as やま, ''空'' as そら, ''人'' as ひと, for example. It was a kind of translation at first.
In Japanese, the meaning of these kanjis, 会, 逢, 遭, 遇, are all あう, but each letter of kanji has difference in meaning.
There is often the most popular and general kanji in the same word in Japanese, it is often indicated in dictionaries. For example, there are several kanjis for あう, ''会'' is the most general, doesn't have a nuance, and the others have specific meanings.
The kanji of 逢 conveys that a man and a woman who are in love meet.
遭 conveys that someone meets with an accident or a bad happening.
遇 conveys that someone meets someone or meets with something unexpectedly.
You are able to express more details with choosing kanji according to the situation, but it is not general to use 逢, 遇, and 遭 in daily communication like E-mail.
Native Japanese speakers, including me, don't care which kanji should be assigned when they say or hear ''あう'' in conversation, because あう is just one word.