Have these things happened in really quick successions...
1) Have Israel as the one who actually launch the attack first, more like quick pre-emptive attack that going too well (So the Soviet Union will be less reluctant to support the Arabs with Nukes, they have policy to use it for 'defensive' purposes. America start the nuclear escalations is nigh impossible because their Presidents actually must cultivate political support with the civilians, instead of bureaucratic oligarchy)
2) Have their pre battle preparations actually reversed as well, this is also Ramadan, so the Israeli attack ends up killing innocent civilians.
3)Israel not only quickly seize control of OTL golan & etc, but press much to Lebanon, and in Egypt, to the other edge of Suez.
4) Have Israel do much better than expected (but with them on the attack instead of defense as IOTL), so they will have enough arrogance to not following the inevitable UN calls for ceasefire.
5) Soviet threaten them with Nuclear Strike if they do not put a cease fire, Israel are arrogant enough not to follow it even after USA threaten to withdrawn their supports (they won't, only baseless threatening as usual).
6) Soviet army reinforcements are sent to help the Arabs, they drop tactical nukes to Israeli troop concentrations...
7a) The rest is history... unless...
7b) US government finally decides they're tired with the Jewish lobby and basically say to Israel "You asked for it". Palestine will be founded as Arab nation, with Soviet later catch a lot of international flak about nuclear weapon usage... I can actually predict earlier, much earlier revolution in eastern Europe, but USSR won't fall outright, just crippled because at least communism will be popular in the Arab / (Formerly) Islamic world.