It's pretty crazy to assume that we wouldn't be around in a millennia, two, or three. No other civilisation has been truly global, no other civilization has achieved so much in terms of mastery of nature and technological advancement.
So unless the carrying capacity of earth suddenly collapses to the point that it's impossible to maintain organised society for several hundred years, then I think we'll be alright. If that does happen, then a few nomads stumbling upon a nuclear waste dump will likely be the least of humanity's problems!
Many times over? I doubt it, if you are talking about nuclear weapons.
There's no way we could punch through all the ice at the poles to get any microscopic life that might live under there, and in a standard nuclear war nothing would touch them anyway. Nuclear winter (if that's still thought to happen in a likely scenario) might not even get all the surface life.
There are also people on this planet that are actively preparing against such a scenario. If anything is going to kill life (as we know it) on the planet it would be humans who just shrug and turn up their AC after a +4°C global temperature change and try their best to go to something ridiculous like +9°C as if destroying the planet was some sort of sport.
I'm pretty optimistic that we can avoid such a scenario but you never know...
A nuclear holocaust would render the problem of some cancer ridden nomadic humanoid type creature stumbling upon a nuclear waste dump fairly moot.
It's still exceptionally unlikely that such an event would take place, given that humanity managed to stagger through the cold war without such an incident occurring. There are no longer any grand ideologies fighting it out for global supremacy. Most countries look fairly similar in terms of economic systems.
A global blight is exceptionally unlikely, the current free trade regime will enable the mitigation of localised shortages as the climate changes and the carrying capacity of the earth reduces precipitously. This will enable humanity to find technological solutions to these challenges as they become more pressing.
The two primary constraints on humanity presently are energy and fresh water abundance. There are a number of promising candidate technologies to address the problems of energy abundances including: advances of hydrogen electrolysis, nuclear power (as mentioned), fusion, wind, and photovoltaics. Increasing energy abundance will partly address the problem of scarcity of fresh water by making desalination cheaper, but there have been a number of advances in material science that have also shown great promise in this area.
If you combine these advances with novel approaches to farming, such as vertical farming, then the carrying capacity of earth could increase by several orders of magnitude!
This is not even to mention the numerous proposals to at least delay the onset of global heating through climate engineering. This delay would give humanity time to adjust to the new conditions. They could even possibly be used to mitigate the problems caused by previous pollution subsequent to a switch to renewable energy while the earth heals as it gradually expels the carbon dioxide from the atmosphere.
People running around today, panicking over a supposed coming apocalypse, remind me of Malthus. You cannot look at the current technological conditions as a fixed factor when evaluating the carrying capacity of the earth. It completely ignores the fact that the planet is inhabited by the most ingenious creatures in the solar system.
So unless the carrying capacity of earth suddenly collapses to the point that it's impossible to maintain organised society for several hundred years, then I think we'll be alright. If that does happen, then a few nomads stumbling upon a nuclear waste dump will likely be the least of humanity's problems!