No. Climate change will be bad but won't render Earth uninhabitable. It probably won't even be as bad as any of the major mass extinctions we know about. WW3 is not assured, and all the major powers understand why nukes are a terrible idea. Even if things go hot between US/CN I bet it'll stay conventional.

Furthermore if those disasters do happen, they will surely kill a martian colony, since it won't actually be independent of earth for decades, in terms of spare parts, genetically viable population, etc, even if they do manage to grow their own food and oxygen.