It wasn't only fossil fuels, but all important energy producing elements were being hoarded and fought over. Imagine our world (irl) but with 3/4 of our energy producing elements (fossil fuels, plutonium, uranium, tritium, etc.) Cheap, clean, energy wasn't becoming perfected till about the same year as when the bombs dropped. Even the nuclear cars (like the Corvega) were still running on gas as well as fissionable material (Personal logs in Dead Money refer to "Chryslus gas hogs" )
With the world population still rising, and "known" energy sources drying up, the resource wars began.