I'm a mechanical engineer by trade and in my day to day life I use complex mechanical elements which today are considered commodity items - roller bearings, screws, flat surfaces, measuring equipment, bar stock, and so on.
Now we learn in engineering school how important flat surfaces and precision screws were to the industrial revolution, and we learn a few of the big players' names like Whitworth, and then a few methods like the 3-plate grinding method for flat surfaces, but the curriculum doesn't go into much detail on the history beyond those brief shout-outs.
I'd like to look more in depth into the development of modern machine tools and their construction elements, manufacturing methods, and how long it took these machines/methods to be commonplace (subjective, I know).
This is a very broad, detailed question. You are talking about not only the innovations in machine tools, by inventors like Joseph Whitworth, Henry Maudsley and Joseph Bramah but innovations in manufacturing that drove them and were driven by them.
For the men and the machines themselves, there's a decent site for biographies , with some useful links, here . To get into the fine details of the machines, there are also fortunately period manuals on machining: some of these , and articles from machinist magazines of the 19th c. and later, have been reprinted by Lindsey Publications. These are increasingly hard to get since Lindsey himself retired, but you can browse old stock at Your Old Time Bookstore. Some of these old machining books are now digitized and available online. Project Gutenberg, for example, has the 1887 Modern Machine Shop Practice, by Joshua Rose and the 1919 Turning and Boring by Franklin Day Jones. Digging through these, you can trace the small improvements: for example, see an evolution from carbon steel cutting tools being standard, in 1887, to high-speed tool bits becoming common, in 1919 ( and shortly after this in the 1920's the Germans would discover and develop carbide tools). If you are in the US, the building of the old Robbins and Lawrence shop is now the American Precision Museum in Windsor, VT. Their collection is worth a look: it includes an early 19th c. metal lathe - with a granite bed ( easier to make a 6 foot bed from granite than cast iron, I guess) .
For an overview of manufacturing, Hounshell's From the American System to Mass Production and Merrit Roe Smith's Harper's Ferry and the New Technology are classics, exploring how reputability in production of arms components spread to the manufacturing of other things...or perhaps I should say, how the drive for repeatability in production of them influenced other manufacturers.
Important to the achievement of repeatability was the development of special cutters, gauges, jigs and fixtures. These don't get enough attention- perhaps because they were, for the most part, disposed of when the product changed and so most don't survive in collections. But you can get an idea at the AP Museum, which has an old Robbins and Lawrence milling cutter that was to profile one half of a lock plate for an Enfield rifle, and the Springfield Armory Historic Site still has some of the old gauges used there in the 19th c. and 20th c.
Also important is the gradual standardization: of things like tapers and screw sizes, in the 19th c., resulting from agreements among engineers, engineer societies. I regret that I don't have a good source for that.