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.

The modern Chinese writing system is fully phonetic, just with extremely complex spelling. There is no pretense that characters represent ideas or words. They represent syllables.

Phonetic use of the characters was immediate. The go-to example here is 來, which depicts a stalk of wheat. It is the spelling of the verb "come", and the verb is spelled that way because the character for "wheat" was borrowed with no alterations to represent its own pronunciation, which was shared with the verb.

My understanding of the history of Chinese writing is that it kept trying to go phonetic, but each time they prevented it, because the writing had to be read across an empire with multiple languages. Even so, something like 20% of the characters are quasi-phonetic, with the radical giving the topic and the rest of the character giving the approximate pronunciation, so "the word for {a plant|a thing of metal|a person|etc} that is sounds similar to X".

When the Japanese imported it, they used the characters much more phonetically. They used the whole word when that worked, but the characters got assigned to the Japanese pronunciation of the word, as well as the pronunciation from the pieces of other words where that character appeared, as well as the Chinese pronunciations. Then six hundred years or so later they imported them again, by which point the characters had evolved in Chinese but not in Japanese. So its sort of phonetic, but it's a complete mess.