The process is fairly straightforward. When there's little enough known about chemistry, biology, and physics that one person could easily be familiar with it all, it makes sense to just call them all Natural Philosophy (as Newton's contemporaries did). When that threshold is crossed, it makes more sense to have specialized Physicists, Biologists, and Chemists.
Timing-wise, it looks like the term "Natural Philosophy" came to be used rather than just "Philosophy" by the time of Aristotle, and continued in use until at least the mid 19th Century. However, you'll find that people didn't specialize in just Natural Philosophy until (as usual) just after the information explosion that came along with the printing press. Likewise by the mid 1800's, while it was still in use as an umbrella term, there were scientists exclusively specializing in the various branches of it. The process on that end would have been more gradual.