It was a pretty big deal. Cicero, that omnivore of philosophical schools, read it with pleasure and to some profit. We may reasonably say that Cicero derived at least some of his thoughts on the Epicurean philosophy from Lucretius, though Cicero was by no means at a loss for useful sources on that score. Later, in the early modern period, Dryden began but never finished a translation of On the Nature of Things into English.

Now, that said, I do want to caution against reading the ancients such that they are leading to some contemporary view. This is often true, but is almost never straightforwardly true. That is to say, it is not a baffling coincidence that certain ancient views are taken as predecessors of contemporary views, but it would be a mistake to infer from this that the ancient views ultimately have all that much in common with contemporary ones.

Take your example of evolution. First, to answer your question about how revolutionary On the Nature of Things was, Lucretius and the Epicureans were by no means the first to suggest the possibility that animals are as they are as a result of blind chance. (Or, for that matter, to be materialists or atomists, with Democritus of course being their most obvious predecessor in that respect). Way, way back during the Milesian period of philosophy (which, incidentally, is about as far back as it is possible to go) Anaximander declared:

In the beginning humans were born from animals of a different kind, since other animals quickly manage on their own and humans alone require lengthy nursing. For this reason they would not have survived if they had been like this at the beginning. (12A10)

There is some hint here of humans coming from other animals, reasoning to this conclusion on the basis of fitness for survival. That is a relevant similarity to contemporary views and to those of Lucretius. But obviously you can see that, while this is so and while we could rightly identify such a view as a precursor of the later ones, it would nevertheless be foolish of us to take the views of Anaximander to be more scientific (or Lucretian) than they actually are. And of course there is the exceedingly famous argument of Empedocles, whom Darwin was later to credit with "shadowing forth" the theory of natural selection. As Aristotle says of this argument in the Physics:

A puzzle now arises: why not suppose that nature acts not for something or because it is better, but of necessity? Zeus's rain does not fall in order to make grain grow, but of necessity. For it is necessary that what has been drawn up is cooled, and that what has been cooled and become water comes down, and it is coincidental that this makes the grain grow... Why not suppose, then, that the same is true of the parts of natural organisms? On this view it is of necessity that for example, the front teeth grow sharp and well adapted for biting, and the back ones broad and useful for chewing food; this result was coincidental, not what they were for... On this view, then, whenever all the parts came about coincidentally as though they were for something, these animals survived, since their constitution, though coming about by chance, made them suitable. Other animals, however, were differently constituted and so were destroyed; indeed, they are still being destroyed as Empedocles says (198b16-34)

So the first point is that Lucretius is not the first to have given this sort of account. But the second, and frankly more important for our purposes, thing to note is that this is not anywhere close, really, to the Darwinian view. Empedocles is not thinking animals are slowly evolving over time to better fit their environment. He seems rather to think that organisms just come to be more or less at random and the ones that aren’t totally unsuited go on living. He also isn’t basing this view on any systematically carried out program of observation; in point of fact, it is to Aristotle that we owe that particular scientific outlook. Empedocles ultimately bases his arguments on the twin principles of love and strife.

Now, interestingly, Aristotle argues against this chance based explanation (198b34-199b1). His basic point seems to be that the results of nature come to be always or usually. Not always, because sometimes things go wrong; sometimes a tiger might be born three-legged. But, says Aristotle, the very fact that it seems right to say that something has went wrong suggests the chance-based explanation cannot be right. Animals are not as they are by chance. Tigers make more tigers, and all the tigers behave in certain characteristic ways insofar as they can. There is something that is what it is to be a tiger, and a tiger succeeds at being a tiger insofar as it is able to do those things. On the chance-based explanation, Aristotle sees no way of making sense of any of this. And neither, frankly, do I, at least if I limit myself to the resources and evidence available to a philosopher like Epicurus, Lucretius, or Empedocles.

Of course, the contemporary advocate of evolution has an answer here in terms of natural selection and genes and so on. But Lucretius, Empedocles, and their ilk have, it seems, no obvious story to tell. Small wonder, then, that Aristotle’s picture wound up carrying the day for so long! Put yourself in the perspective of an educated person in 1200 AD who is reading through Aristotle’s Physics. Which view are you going to find more compelling? That animals can move themselves around intentionally, see, hear, smell, and give birth to others of their kind as a result of their ultimately being nothing but atoms bouncing around in the void at random? Or that animals have certain natures that can be described and grasped by careful observation of their behavior, and in virtue of which they are the sort of animal that they are and do the things that they do? I’d wager you’d prefer the latter, especially when Aristotle had volumes upon volumes of meticulous observations, results of dissections, and other experimental data regarding animals to offer as well.

The upshot here is that the reason the modern theory of evolution is convincing in a way that the Lucretian or Empedoclean pictures are not is because those ancient theories, ultimately, bear little more than a passing (if nonetheless interesting) resemblance to the contemporary account. The scientific methodology and specific physical mechanisms posited (included some, like forces, that would seem to be disallowed by the Epicurean worldview) within the contemporary picture, as well as the mountains of scientifically gathered and curated data that have been gathered to support it, make the difference here.