It doesn't help that the Saudis are just one large money-transfer away from having their own nukes (the check is probably already written, just waiting to be cashed, so to speak). The Iranians are closer, and Pakistan's been there for some time. So even if the "grown ups" (USA, Russia, China, India, UK, France, Israel) can be counted on to behave themselves, the upstarts less so.
It's also an open question how well the new members of the club would be able to keep their own arsenal under lock and key. If the Middle East gets into a nuclear arms race, to protect themselves from each other, that's all the more opportunities for some enterprising jihadis.
In one sense, the presence of nukes has made large-scale war more unlikely, and taking them off the table might have the opposite effect.