A general outline of the war is Egypt and Syria catching Israel off guard, and making major headway initially. Israel is fighting back but things are tough. Israel cables the United States and says if they aren't resupplied they're going to have to resort to nuclear weapons, and indeed begins to mobilize them across the country. The United States sort of takes its time to resupply Israel but does it before they use nukes. Using resupply, Israel turns the tides of the war, pushing back Egypt and Syria and almost encroaching on Cairo.
The biggest place for a fuckup is that after Nixon and Kissinger asked the US military to begin sending supplies immediately, the military took its sweet time to do so. It was either 4 days or a week wait. If the military took a little longer, its probable that Israel would have used its nuclear weapons on Egypt (probably the Sinai actually) , which would prompt more Soviet involvement, which would prompt more US involvement, which would lead to the scenario you're looking for.