Newton is not a "typical" example, in that he largely did not publish his ideas until much later in his life, and the popularization of them was largely done by other people much later (the French philosophes in the Enlightenment in particular).

But let us take a more typical example, such as, say, the work of Robert Boyle in the late 17th century. Boyle's work was spread in three major ways:

1. Word of mouth and letter. Early modern "scientists" (the word had not yet been coined) initially spread their work primarily through correspondence with other scientists, or talking to them in person. It might be direct (Boyle writes a letter to someone else about his work) or indirect (someone else writes a letter to someone else about Boyle's work). But there was a "network" of correspondence (to use a modern metaphor) between "men of letters" in this time, in England, France, the Netherlands, the German republics, and Italy, among other "hubs" of research.

2. Demonstrations and work presentations. In several countries, there were institutions established so that scientists could demonstrate their work and present papers to their peers. In England, Boyle was a member of the Royal Society of London, which held regular meetings at Gresham's college to discuss matters of "natural philosophy." This were attended by both other scientists and also interested members of the aristocracy. This is a place that Boyle could present, for example, his work on his air-pump and his theories of the vacuum, and get immediate feedback. He might not be able to actually perform the demonstrations publicly (the machines, etc., could be tricky), but he could present testimony from other scientists who he had shown it to previously, who would vouch for his accuracy. The notes and papers from these meetings, as an aside, were written up and printed, and distributed to "corresponding members" of the Royal Society by Henry Oldenburg, the secretary of the Society. This is considered the precursor of the modern scientific journal, and was a way for people who could not make it to the meetings to be part of the knowledge-production. Those who could not attend could also write letters to Oldenburg, which would be read at the meeting (and inserted into the record).

3. Books and pamphlets. Book publishing was not as cheap as it would become in the 19th century, but it was a lot cheaper than it had been previously. Boyle published numerous monographs, such as The Sceptical Chymist (1661), which contained his research results from long investigations. He could publish, using relatively fine printing techniques, technically detailed depictions of his experimental apparatus, for example, which served both to try and help other people replicate it (this was often not possible, however, because of the idiosyncrasies of this "one-off" tools), but also to give a real sense that these things existed and that his testimony as to their behavior was true ("virtual witnessing," as scholars call this today). Most of the "published work" of scientists from this time period came in this form — long books or, sometimes, pamphlets (e.g., Galileo's Starry Messenger is essentially a pamphlet describing his research results).

Note that in this time period, none of the "news" of their work would have been spread through what we would call a newspaper, though such publications (as pamphlets) were starting to come into existence. This would have been a world limited primarily to scholars. It was not until some time later (again, early Enlightenment period) that you would have expected the diffusion of knowledge to be somewhat more "popular," but even then we are talking about a relatively limited class of people. The true "democratization" of publishing in general, much less scientific publication, did not really occur until the 19th century (so Darwin's context was very different than Newton's — people would read Darwin's book on their train commutes, whereas nobody is reading Newton's Principia Mathematica, in Latin, on their way to work).

As for how long it took — I don't have a number right at hand, but in the early modern period it could easily take several weeks for mail to reach other lands. It was no easy thing, even though it happened. Whereas by the mid-19th century, postal service within England was swift enough that Darwin could send a letter to someone in London in the morning and get a reply back later that day.

On Boyle, a useful book is Shapin and Schaffer's Leviathan and the Air-Pump, which, aside from being about his work, is also about how this "world of letters" functioned in the 17th century. As a contrast, Browne's Charles Darwin, vol. 2: The Power of Place, is all about the rapidity of communication and publication that was possible a century later.