In the US, and likely elsewhere, invasive species would take over for the next few thousand years.
Around me in NJ Japanese knotweed is filling in verges and undergrowth, and there are many bamboos stands that would spread. The local animals don't eat these, so either they'd have to develop a taste for them or find their favored foods crowded out. Bird species would disappear without human feeders.
Of dogs and cats, the hunting and working breeds would thrive, eventually coming into balance with whatever prey species survive, along with bears and wild cat species. Those breeds that require human aid to live would be gone within a few years.
Rats? I wonder if how well they'd survive without humans to feed off.
Raccoons have dextrous paws, are omnivorous, and can be pretty smart, so they might have a chance.
Longer term the once-ubiquitous forests would fill in suburbs and farming fields, though invasive insects like the asian longhorn and spotted lantern fly will reduce diversity short-term.
One factor would be the large quantities of human-generated chemicals, many of which have effects on various organs, along with nuclear facilities. Beyond the massive amounts already in the water and soil, over time the containment vessels for the remainder would break down and release them, generating localized mass death and/or mutation events. Chernobyl is a running experiment in the effects of nuclear radiation on the natural environment.
One article I read some time ago estimated 200,000 years for all surface signs of humans to disappear. There would be a sedimentary layer left behind, many artifacts buried will survive in some form, and up in space the higher orbiting satellites, interplanetary probes, and human artifacts on the Moon could persist much longer. Those future species might even find that layer a source of materials.
Who knows, some day in the distant future an evolved rodent could come upon a long-buried Mickey Mouse phone and wonder if that was one of her ancient ancestors!
This is a relatively recent (updated) article, which might be of interest:
There's a thriving population of radioactive animals that have taken over the abandoned Chernobyl exclusion zone, even though the area is toxic for humans
April 26, 2020
[This is an updated version of an article originally written by Courtney Verrill].
On April 26, 1986, reactor number four at the Chernobyl nuclear power plant, located in the then-Soviet Union, experienced a power surge, resulting in an explosion that sent a cloud of radioactive materials across parts of Europe.
It was the world's worst nuclear accident. Around 350,000 people were evacuated following the explosion.
Today, the areas surrounding the Chernobyl nuclear power plant are almost completely void of humans, save for a number of locals that reside in the Chernobyl exclusion zone.
But the contaminated area is now populated by a diverse wildlife community.
Scientists and researchers are still investigating how exactly the animals are affected by radioactive exposure, but many studies so far point to the most likely explanation for why the animals are thriving: the lack of humans.
"Nature flourishes when humans are removed from the equation, even after the world's worst nuclear accident," Jim Smith, an environmental scientist who has studied life near Chernobyl, told National Geographic.
Here's how nature has reclaimed the contaminated land.
We are the most adaptable species on the planet, which is why we are running it. We'll be here for a very long time yet. And since it seems the doom mongers among us are reluctant to breed, then eventually they will die out, leaving us stronger. Assuming Darwin was right.