How do you think the natural environment will fare?

It largely depends on how you define nature.

Many species of animals and plants are severely threatened and most of them will die out or will only be able to survive in zoo's or wild parks.

Humans at the moment should be considered as a single population due to relatively small genetic diversity and frequent contacts with other humans from across the world. This implies that humans as a species are pretty susceptible to pandemics. COVID-19 is a pretty harmless pandemic, we have known pandemics for other species that (almost) completely wiped out that species.

If you are simply worried about all life on earth disappearing, that seems extremely unlikely. As Stephen Jay Gould said every age on earth is the age of bacteria[0]. Bacteria have always been the dominant form of life on earth. They were the first ones and they will be the last ones.

[0] https://www.washingtonpost.com/archive/1996/11/13/planet-of-...

Most current signs point to the hypothesis that humans are now and have been for thousands of years driving a major extinction event. Even disregarding the impacts of climate change, humans are far from reaching the crest of their impact on the biosphere. With climate change accounted for, the impact of our actions today will be compounded into the future.

Life, if it survives us will likely be pushed through another very narrow gap as it has at least 5 other times in the past. In the very very long run, life may be able to rediversify but, even if we significantly decrease our impact through extreme modification of our behavior or through incredible leaps in technology, we have already been a calamity for life on this planet.

The biosphere would take tens of thousands, hundreds of thousands or even millions of years to recover.

With harvest records being broken almost every year, famine declining, Earth greening with deserts shrinking and reductions in record low temperatures with increasing average temperature, it will do better and better. Humans will thrive, but if one species expands some other species will have to give in. I care about humans more than other species. The future of humans is bright, if we decide it shall be.