There's always a sliding scale between "proto-writing" and fully developed writing systems, but just using symbols phonetically instead of semantically happens much faster than that. The very archaic forms of proto-cuneiform is from about 3400 BCE (though things like clay tokens are much older). It, like virtually all writing systems in the world, developed into a true writing system by use of the "rebus principle", where symbols came to acquire different meanings based on phonetics in the same way as in a rebus. Like, in English, if you had a symbol for "female sheep" (a "ewe"), you could start to use it to signify the word "you", even though there's no semantic connection.

The earliest evidence for this in cuneiform is from around 3200-3000 BCE. There is a famous tablet where the symbol for "reed" is used to represent the word "reimburse", because they're both pronounced like gi. By a few hundred years later, cuneiform was a fully fledged phonetic writing system.