Historic ideas about what "nature" was and how to relate to it are very different from modern ones. The predominant (though not only - see Virgil) cultural view in Roman antiquity, drawn heavily from Greek philosophical sources, was that natural forces (natura) and human forces/civilization (cultura) were complimentary principles or processes. Nature was not an entity or a place - nature was a harmonious and orderly process of growth and transformation which takes place within the world. Human intervention, in the form of agriculture and cultivation, was seen as improving not just material produce of the land, but working with and improving nature and the natural processes. As far as I know, Romans had no particular concept of a "forest" per se - the wood forest itself comes from a Latin word referring to a form of wooded estate managed for game, not a forest wilderness. Greeks and Romans did preserve certain natural places as sacred, but I don't know enough about greco-Roman religious practices and archeology to say what it meant to a roman cult to protect/preserve a sacred glade or spring.

Romans did have a sense that natural spaces (as they understood them) needed to be cared for - roman writers were very concerned with maintaining the health, condition, and fertility of their agricultural lands and woods. Even while overall roman agroecological practices proactively disrupted and destroyed biodiverse habitats like wetlands and montane forests, they still had a core sense that nature could be spoiled by human misuse, and that virtuous and responsible citizens and people managed land judiciously and in harmony with the principles of natura. It doesn't seem like the Roman elite had a concept of extinction or extirpation - while they did notice and write about changing environmental conditions and the scarcity of fish and game, they don't talk much about the disappearance of lions, hyenas, and leopards from Europe, or quintessential "African animals" like zebras from north Africa, though all those events occurred on their watch, or connect the scarcity of animals with persistent over-exploitation.

The middle ages were different, and more of the same. Almost everything we know about medieval environmental thought isn't particularly environmental - it's theological. The medieval (Christian) elite had an important conception of "wilderness", inclusive of places like old growth forest or rugged mountains, which is largely synonymous with "wasteland" or "badlands". This elite was concerned with the moral and spiritual quality of the wilderness and it's relationship with humanity and Christ. Wilderness was not 'useful' (to your original point), nor safe, and was typically conflated to the absence of God. Monastic writing in the middle ages might more often use "forest" to mean a managed woodland or park, while a forest would often be called a "desert"! So here we're again not seeing the idea of a pristine "nature", or the importance and virtue of wild things and wild places. Medieval writers do find virtue in harmony with nature, and godliness in the order and growth of things, similar to Roman thought, but you also see natura appear as an entity - often a woman or nymph - variously wild and dangerous or benevolent and harmonious. There were other trends in Christian thought, and the high middle ages saw a rise in transcendental religious philosophy which often embraced a more expansive and not always benevolent conception of nature as a manifestation of the power and ineffability of the divine, and we do have some practical writings on forestry and game management, so there is some variety in the thinking of the time, but none of it really speaks to the sort of nature for nature's sake thinking you're talking about.

Whether any of this actually represents a common cultural understanding of nature in the middle ages or just represents a literary tradition among literate religious elites that existed in contrast to a pagan/folk socioecology is anyone's guess, though I've seen some speculation and anthropological gesticulation on the subject. Either way, nobody seems to have been much concerned with the "preservation" of forests or habitat - for one thing, there weren't any old, mature forests in the Mediterranean, British isles, or much of Iberia or France to speak of, and in central and eastern Europe old growth forests were old because they were beyond significant human influence until later centuries. All of that said, historic people generally found a way to live with nature, and had their own concepts and practices of sustainability. Because of all the factors above, medieval European sustainability and traditional ecological practice would not look much like what we might think of as sustainability or "preservation", but is more akin to modern conservation and land stewardship.

Everywhere in Europe, humans adapted and adapted to their ecosystems, and truthfully, almost every "natural" place you can find in Europe today is either less than a hundred years old, and/or a product of thousands of years of co-existence with humans. Traditional coppicing and sylvopasturage produced sustained landscapes of open woodlands which favored certain human and wild uses over others. While sometimes haphazard, restrictions on hunting, fishing, logging, foraging, and other resource management practices seem to have existed to preserve the landscapes themselves from being spoiled, as well as the resources within them. Slash and burn practices were used to create nutrient rich fields out of forestland, which would then be fallowed, then pastured, and then allowed to succeed back into forest before being burned again by another generation down the line. There was some understanding that the regenerative nature of this cycle was important, because we have writings with people complaining about greedy or short sighted neighbors not fallowing and succeeding their fields in favor of trying to maximize their arable land, but I still wouldn't conflate medieval thought or natural resource management practices to "preservation", and while some may not have been for "direct" human use, human use was still unequivocally a motivator.

Preservation and conservation as such don't really seem to show up at all in western thought at all until the 17th and 18th centuries, and there as a response to the very obvious and ongoing ecological crises of deforestation and land depletion arising around the 16th century. Beyond that, early environmentalism doesn't really appear at all until the 19th century, surrounding the events and concepts you referenced in your question, and in part borne more from romanticism and the desire to return to a semi-fictitious "nature" than actual ecological practice. Some things, like the establishment of the national parks, weren't even that - conservationists like Roosevelt wanted there to be national parks and game preserves very specifically for direct human benefit.

I hope this helps - most of the details from this answer come from Richard C. Hoffmann's Environmental History of Medieval Europe, but if you have other questions I may be able to help point you to other sources. Unfortunately I can't answer this question for anywhere outside of Europe or some of North America. If there are robust environmental histories of southwest, southeast, or eastern Asia, they aren't in English or I haven't found them, though there is a lot of new historical agro-ecological research coming out of China and India these days.