Lots of reasons. I'd start by asking "Why not later?" in return, because if we start by just focusing on the first industrial revolution, then what most (reasonable) literature^1 will agree on is that it's all pretty complicated, and the conditions in which it occurred were quite special. Nobody is 100% sure which of the conditions it happened in are actually causes and which are non-causal (correlations), and to what extent. But some of things often credited (correct or not) for the first industrial revolution are:

The laissez-fair government attitude in Britain, other government policies, the well-developed academia, the scientific method, the international trade and competition, war, geography and deposit locations, agricultural innovation, dependence of an island nation on agricultural imports, colonization/exploitation, urbanization and overall population level, religion, culture and other social aspects, and last but not least the actual hard/physical technology already present.

And that's just what I can think of of the top of my head; there's more. Of these colonization is -I think- the most disputed topic. There's historians who argue that the colonial empire was crucial for allowing Britain to develop new industries by through things like amassing capital and relieving the island of certain production burdens, and on the other extreme there's historians who argue that the use of underdeveloped, low productivity lands actually inhibited industrial development as well as being an enormous burden on the government. We'll probably never be able to put into numbers how much of a cause any of these were, but that's okay because it's history, and the situation was completely unique and it's never going to happen again. There's plenty of patterns that can still be discerned or argued for.

So there was a very large foundation for the first industrial revolution to happen. And this foundation had been centuries in the building. And this is the pivot of the answer to all your questions I think. The industrial revolution is called that: a "revolution", yes. And it was revolutionary. But the 16th and 17th century before it hadn't exactly been static either, nor had the middle ages or the ever-depreciated dark ages!^2

So what was required and why didn't the ancients just do it? Well, for many reasons: For one, the oft-mentioned trade institutions were completely undeveloped in ancient Rome and Greece as compared to the 18th century. This includes not just to trade laws and tariffs, but also to crucial banking and credit methods that just did not exist at the time and wouldn't exist for over a millennium. The 'soft' inventions of investments, lending, stock exchanges, insurances et cetera has made a huge impact on how production was done and how both companies and countries could function and grow. Most of these institutions precede the revolution by quite some time, as pretty enormous companies and factories were already established all over Europe long before the first train rode.

I deliberately say factories, because that's what the large textile plants and extensive ironworks by all means were. What comes to mind with the word 'factory' nowadays might be a giant reinforced concrete hall with a very tall roof, lots of conveyor belts and immense work distribution. But that's really 20th century stuff. The factories that arose at the end of the 18th century weren't immediately all that different from what was already present, which were the many industries that had been slowly but quite steadily increasing the productivity and raw output of each worker over the decades. Distribution of work and mass production had not really been properly applied up to the industrial revolution, but they weren't applied much in most of the 19th century either. The same can be said for myriad measures that we associate with industrial production now and seem obvious to us, but had to be developed over time then.

I'd like to focus more on the mechanical/physical aspect of the industrialization and machinery, because I like it more, and because you also basically asked why they didn't "just mechanize already, with the iron and copper they had". Well, put simply, it's because there's a couple hurdles:

1. Building it. You need much more than basic iron working to build a practical steam engine, or other post-1800 industrial machinery.

2. Use. Just having a practical steam engine, isn't all that useful on its own.

3. Maintenance. A higher-tech industry requires a higher-tech industry to support it.

To start with #1, it just so happens that /u/half3clipse wrote a little on this last week. He mostly mentions the lack of theoretical basis for the design of a practical steam engine, but that's not even all there is to it. Ancient metallurgy was also multiple steps behind that of the Renaissance and beyond. It's ironic how the popular image of the Middle Ages is a knight in shiny full plate, and also that the Middle ages were more backwards and stagnant than the Classical period, yet this (late) plate armour was metallurgically vastly superior to what the best smithies in the Classical period could produce. Now it's not actually impossible to build some sort of low-pressure steam engine with shitty iron, but it's going to be less efficient, less durable, much less powerful for the same weight and size, and more costly if your society hasn't developed the smelting techniques that allowed iron and steel to become stronger and cheaper at the same time. All that alone is enough to practically ruin your chances, not so much of having such a machine invented, but rather especially of it being used in a reliable and productive way.

Put differently: There was nothing literally stopping a hypothetical roman metalworker from grabbing a ton of iron, casting or shaping it into a set of rails and wheels, and produce a more energy-efficient transport system using a horse-pulled train to get iron ore from a mine to a smelter. Except... he'd be using iron that was horrendously expensive at the time, using many more work-hours per meter of track than an industrial worker in 1800 would, to produce at best a rails that wouldn't last for more than a couple days of use... and finally the actual economic gain/productivity increase of the mine-smelter complex would be so low that it's entirely conceivable that it wouldn't even make up for the investment of the material and man-hours. Even with the idea or execution there, there were no blast furnaces around, nor were the shaping techniques and tools to get it done in a profitable manner.

#Part 2 below