There isn't any single answer to this, because there isn't any coherent definition of "science" that actually encompasses everything that people mean when they say "science." (There are non-experimental sciences, non-observational sciences, sciences that deal with unfalsifiable theories, etc.) This is an extension of what philosophers of science call the "demarcation problem" (there is no clear demarcation, or line, between science and non-science), which has basically been agreed as being unsolvable in a strict sense, and applied backwards it means there isn't going to be some way to say, "ah, now they are doing science, finally!"

The best answer one can give, then, is when people started saying, "we are scientists and we are doing something different than philosophers." Even that's complicated (people started thinking of the work of science as being distinct from, say, deductive philosophy long before they had the word "scientist"), but you can more or less say that the investigative ethos of science as we understand it to be in its modern sense starts to become more articulated and widespread in the 17th century (the "Scientific Revolution," though historians don't like that phrase either), where you start to see more attention to things like the uses of experiment as a means of finding truth, the use of peer review and other institutions of collective knowledge, the rising prominence of specialized instruments that extend beyond the human senses, the push for a search for "laws of nature," and the increased use of quantification as the language of science. Even here, there are plenty of exceptions, but this is more or less when you start to see these ideas becoming more widespread amongst people who thought of themselves as really investigating nature and seeing that as different from most of philosophy. But they still called themselves natural philosophers in this time, which is interesting and complicating again!

What we consider to be modern science, with its terminology ("science," "scientists") and institutions develops in a more fully-formed mode by the end of the 19th century, with the full "professionalization" of science (it becomes a more regularized career with standard expectations of educational pathways, etc.), the emergence of the research university, and other things that make it look a lot like our present-day world (even then, many aspects of our modern scientific world don't emerge until the post-WWII period).

Which is to say: there has been a lot of shift over several centuries, and what we think of as "modern science" is relatively new (only 150 years old or so). Over this same time, what we call "philosophy" itself altered quite a bit as well, often distancing itself from the interests of the scientists (but not always). None of this happens in one swift moment (hence historians' unhappiness with the term "Scientific Revolution," which implies a greater and quicker break from the past than is warranted), and you can find people on all sides of it who embody the opposite notions (you can find people who essentially look "scientific" well before the "Scientiific Revolution," and you can find people who really don't who are taken seriously — say, Sigmund Freud — well after it). The "terminology" of "science" and "scientists" did not really become common until the 19th century, which is probably well after the intellectual developments really began (the terms lagged the changes in the field, as often happens).

TLDR; — like so many things in history, it's complicated! If you wanted to say "the process seems to start around the 17th century and was basically complete by the 19th century" I wouldn't fight you over it, but that glosses over a lot of complexity and historians could point to all sorts of exceptions to that generalization.