If Israel does not get hit by the Soviets, which it may, IMHO it will survive at least in the short term. When the big boys wind down, they have nukes and can eliminate any organized Arab opposition. Egypt just needs on detonation - take out the Aswan High Dam, and the flood will do all the hard work. Damascus and Baghdad go away. Israel will be able to seize the Sinai, and the real although limited oil resources there, and can take the canal if they want, both sides or just take the east bank and use it as a moat. The only issue in Syria is can the Israelis grab enough territory that is readily defensible and has the oil fields, otherwise why bother. Israel can find friends among the Druze, Lebanese Christians, and the Kurds - they can all have their own states in alliance with Israel. With the exception of Iran, none of the Arab states have much industry to manufacture even small arms let alone tanks, aircraft and spare parts for same. A lot of maintenance for the more advanced weapons in much of the Arab world is done by foreign contractors. Iran is potentially a major threat, however in the wake of the war their economy will be in bad shape (no export of oil) and if the Israelis want they can use a few nukes there to destroy the key industrial infrastructure as well as the leadership.
Israel has a highly educated population, is pretty much self sufficient in calories, although variety would suffer, and has the most robust industrial base in the Middle East. If the Soviets toss even two or three nukes there, they are toast. Given the prevailing winds the fallout from NATO targets should not be too bad, of course a shift at the wrong time could be a problem, although Israel has a robust civil defense system.
Most of the Middle East will descend in to internecine warfare. Christian-Muslim, Sunni-Shi'a, Arab-Persian are just the start. Larger groups like Kurds and Druze may have a shot, smaller like the Yazidi unless protected by a larger group are going to die. And don't forget the famous saying in the Arab world: "me against my brothers, my brothers and I against our cousins, us and cousins against the tribe, our tribe against others".
Anyways, as stated absent Soviet hits or being blanketed by huge amounts of fallout Israel will not be overrun by its neighbors. If, for whatever reason, Israel is prostrate, the Jews who survive the invasion will envy the dead.