I think Israel (and, unlike GP, also Iran) are too small-time to start real global war. They're more the potential Bosnias or Serbias - small, belligerent countries that could drag their patrons into conflict.
More than enough to bring the Fallout series to life.
Subsequent nuclear weapons projects in erstwhile NPT NNWS could easily trigger a few regional wars, and from there the escalation to a global conflict is certainly possible.