I can only say something about Chinese. According to my experience, when you are a beginner, the learning of the spoken language is quite independent of the learning of the written language. That is, one needs to spend a certain amount of time to learn the spoken language. Of course, ability to read Pinyin can help with the spoken language, as it lets you read dialogue transcripts, etc., but I don't see any way knowing the characters at this stage can have any influence on the progress with the spoken language.
The influence in the other direction may be a bit stronger. If you are completely lost when it comes to differentiating different aspects of syllables (initials, finals, tones), then it's probably too early. For a beginner, it's hard enough to build an association between, say, 車 and che1, remember its tone, and distinguish it from similar-looking characters, such as 事. But if you additionally can't hear or can't pronounce the difference between, say, che1, chi3, qie1, zhe4 and/or other such syllables, then building associations between characters and their pronunciation will be much harder.
So I would say the lowest level when it makes sense to begin learning the characters is when you have internalised the basic phonology of the language and can pronounce most of the important distinctions (and hear them, at least when spoken slowly and in isolation). I'm not saying one needs to be perfect, but the ability to distinguish the sounds must be much higher than by chance.
How about the highest possible level? I would say that apart from these initial prerequisites, character learning, again, is to a large degree independent from learning of the spoken language. A large part of character learning (identifying the character shape, mapping it to the pronunciation, remembering the strokes of a character) doesn't depend on the knowledge of the spoken language. I don't see reasons to believe that one's spoken level can influence how much time it will take. I know from my experience that it's possible to achieve a level around B1 in Chinese, using audio courses such as FSI Chinese, ChinesePod and private tuition, with minimal character knowledge.
John Pasden, an SLA linguist and a Chinese teacher working for ChinesePod, said in one of the podcasts that B1 (intermediate, in the ChinesePod terminology) is also the highest level that you can get to without knowing any characters, and I agree with that. Even if you're solely interested in the spoken langauge, at higher levels you simply need to understand expressions that mainly appear in writing, and it's very unlikely you can simply pick them up by immersion. Without knowing characters you will lose a lot of the clues about meaning of unknown word. Meaning of an unknown word can often be deduced from the meaning of the characters it consists of. But if you only know the word's pronunciation, you have a much harder task, due to the huge number of homophones and near-homophones in Chinese.
To sum up, I believe one may start the writing system of Chinese pretty early, basically just after getting familiar with the phonetic system of the language. One may also start learning it very late, but probably not later than after achieving the B1 level in the spoken language. The time required to learn the written language should take roughly similar time and effort, regardless of whether one starts early or late. Of course, there are several factors that influence that time, but we should remember that one of the strongest factors in language learning is motivation.
So my advice would be: start learning the written language whenever you feel like it. If you want to start early, do it, and if you want to delay it and concentrate on the spoken language, that's also OK. Just keep in mind that you'll need to do it at some point, as long as you aim for a level higher than B1.