Brian Cox is correct that we are likely 15-20 years from having thousands of people living in space on near Earth asteroids, space stations, the moon and on Mars. However, he is talking about achieving a fingerhold in space as guaranteeing the immortality of our civilization. Space mining and space colonization would be the beginning of making humanity and human civilization significantly more resilient.
Cox mentions that the planets and asteroids offers a billion times more mineable resources, including valuable heavy metals such as nickel, cobalt and gold.
Brian Cox seems to mainly be concerned about resource depletion and ruining the climate of the Earth and extinction level wars.
Resource depletion, global warming climate issues and nuclear war are not events that would destroy all life on Earth or destroy human civilization.
Fossil fuels will not be depleted even if it takes 200 years before we transition to solar, wind, hydro and nuclear power and electric vehicles.
Climate change will not cause billions to starve or break civilization. There are geoengineering options for simulating the effect of volcanoes that will prevent the various worst scenarios. However, localized famines and regional upheavals are things that should be avoided.
Direct war between the USA and China or the USA and Russia or other great powers will not happen.
The current worst case wars over the next couple of decades are :
North Korea war with the West and a couple of dozen nuclear weapons used.
Iran war with Israel and the West. This could involve nuclear weapons anytime after 3 years from now.
India and Pakistan nuclear conflict.
Those are to be avoided and could each involve millions to possibly tens of millions killed.
Living on Earth, moon, Mars, and space stations with solar system resources and non-fossil fuel energy would remove resource depletion as an issue unless individual power and resource usage escalated millions of times. There would be redundancy in the resources and multiple independent biospheres.
Wars involving space technology between two or more technology advanced groups would be more destructive than even any war involving just our current nuclear capability.
If there is no FTL (faster than light) travel, then having colonization starships traveling in different directions at over 50% of the speed of light would be effectively unkillable.
An expanding sphere of interstellar colonization would reach a point where any one threat would not be able to reach all parts of the civilization.