Well, to try to take this forward, say we assume that, say, King Hussein of Jordan doesn't make his final attempt to warn the Israelis on the 25th of September, so they don't listen to the final warning they received on October the 5th, and so don't start mobilising.
This leads to heavier initial losses, and that, combined with their complete failure to correctly predict the Arab invasion leads them to panic, and to go nuclear.
Now, I strongly doubt that the Israelis are desperate enough to make strategic strikes, but lets say they make tactical strikes on the rear of both the Syrian and Egyptian forces, followed up by immediate counter-attacks on the front lines.
What happens next?
I very strongly doubt that the Soviets will retaliate on behalf of their clients, although they will have to so something to retain credibility. The threat to supply nuclear weapons to Egypt may well be made, and is likely to be decisive one way or the other.
The oil embargo will become absolute and the price of oil will shoot up even more than iOTL.
This will be very damaging to NATO. I expect the UK and France to completely disown Israel, and put significant pressure on the US to do the same. They have no interest in being annihilated thanks to what they see as a rogue US client..