From the first sentence of your question, I think you are asking "What is the earliest musical score that a musician of today could easily read?"
Consider an analogy with writing: Have you ever seen a manuscript by Shakespeare? (I think there exists exactly one.) It is written in English and in the same writing style as his contemporaries, but it's not easily readable to a person of today. The letter forms are very different. You can puzzle it out one word at a time. A researcher who reads lots of these old manuscripts can read it fluently.
The earliest writing we have is clay tablets we can't read. The earliest writing that we can read (i.e., that some professor somewhere can decipher) is such-and-such. The earliest Old English writing is such-and-such. The earliest Modern English writing is such-and-such (something like Shakespeare's manuscript). Etc.
There are ancient manuscripts that we know are music but we can't decipher. There are ancient Byzantine musical manuscripts that experts can decipher.
Gregorian Chant from around the year 600 is the earliest music in a notation similar to modern notation (notes on the lines and spaces of a staff with a clef). This notation is still in use in the Roman Catholic Church. Music from Shakespeare's time looks funny with its dots and points and diamond-shaped notes but you can read it. It's called "pricksong" from the dots and points.
Is a J.S. Bach manuscript an example of modern musical notation? That's just a matter of opinion. Bach uses obsolete clefs and sometimes beams the notes differently from how we would.
So I think the answer to your original question is "sometime during or after Shakespeare's time and before Bach". That would be the 16th or 17th century, or late Renaissance / early Baroque.