As a kid growing up in Egypt, we obviously had things in our official history textbooks that said that we "totally won that war", and I'm very sure that it's the exact opposite in Israel. So please, I want the answer to be completely objective. I want to know what would have been the logical outcome based on the forces at hand for both armies at the time. Would the destruction of the trapped Third Army by the Israelis prove to be the winning move? Would the Soviets have gotten aid to Egypt in time?
In Military terms, the surrounded Third Army would have had to surrender soon. Egypt had no reserves on that side of the Canal, so a drive north to the coast would have cut off the other Armies as well. This would leave the major cities along the Nile open to attack and occupation.
Adding a little political insight, aside from major power intervention, Israel has such a small army that even this kind of victory would be too great of a strain. They would have to guard these prisoners and feed them, and any advance would require them to do the same to a hostile populace. They really didn't have the ability to do this for long, if at all. It would be like having a tiny snake swallow a watermelon.
So the quick armistice was a relief to both sides. Egypt did not have to see their army taken apart as in the Six Day War, and could console themselves with a very well run set of battles in the Canal Crossing and initial engagements that hurt Israel badly. Israel got the quick peace it needed and regained some confidence lost by that first engagement. Both sides learned that war can be a chancy thing, which eventually got them to the peace table at the end of the decade.
As for major power intervention, aside from the absurd idea of using strategic nukes on cities, the Soviets in particular had no way to change the force balance on the ground in time to do the military situation any good. Remember, it took 5 months for the US, a nation far more equipped to ship troops worldwide to gear up for Desert Storm. Where the Soviets could and did put forth pressure is to get an armistice quickly, which as I said was really in the best interest of all the players, even Israel.
It's hard to say what would have happened had the UN cease fire not happened, but the military situation on October 22nd greatly favored the Israelis on all fronts.
On the Syrian front, things were not looking good for Syria. Israeli troops were 40km from Damascus. The Israeli air force had superiority and had knocked out the Syrian air defense system. Israel was focused on stabilizing the front to concentrate on Egypt. Syria had been resupplied by the Soviets and was preparing a counter-attack and were bolstered by an Iraqi expeditionary force, however their track record against the Israelis did not bode well.
In the Sinai, both sides had crossed the Suez Canal, but the Israelis posed the much greater threat. The Egyptians would have to fight across 100 miles of the poor terrain of the Sinai to threaten Israel. With few roads, little cover, and outside their umbrella of SAM protection, the Israeli air force would have a field day.
On the other hand, Israeli forces were just 40 miles from Cairo and able to threaten Egyptian cities. Egypt had lost large numbers of armor and aircraft, and their vital SAM umbrella was being cut to pieces by the Israeli cross-canal counter attack. Their units in the Sinai were in danger of being cut off and the Israelis were preparing to reduce these pockets. The Israelis were about to take Ismailia, a major road and supply hub for the Egyptian units fighting at the canal.
At sea, the Mediterranean navies of both Egypt and Syria were defeated. The small Israeli navy had the run of the coastline, bombarding installations and landing commandos at will. The Gulf of Suez was blockaded damaging the Egyptian economy. On the other hand, the Egyptians had blockaded Israel at Bab-el-Mandeb beyond their reach.
In short, Israel was no longer surprised and had the upper hand on all fronts and directly threatened the heartland of both Egypt and Syria. While they were fighting a two front war, they had the option of switching to a defensive stance. Militarily, the UN cease fire saved Syria, and particularly Egypt, from more defeats. What Israel would have done had they actually conquered significant amounts of Syrian and Egyptian territories... well, they don't manage what they have very well as it is.
The wild cards included whether Jordan would intervene on behalf of Syria (post war documents suggest nothing more than a token), whether Israel would be able to keep up the pace in a two front offensive battle, and whether both sides would continue to be heavily supplied by their superpower sponsors.
It is difficult to say "won or lost" in a war like this. Egypt and Syria can be said to lose the war because they failed in their objectives in starting it.
What the ultimate outcome might be is incalcuable because both the US and USSR were threatening to intervene. At that point it is not Israel vs Egypt anymore, it is World War III, a war which the USSR could not possibly win and could have ended with the annihilation of their culture.
As far as the Mediterranean situation was concerned the Soviet block was at a huge disadvantage. The US and Britain have multiple bases in Italy, Turkey and in Gibraltar, compared to a single, small base for the Soviets in Syria. Strategically it was a losing war for the Soviets.
The Israelis demonstrated their superiority over the Egyptian and Syrian armies repeatedly, so as long as they continued to receive fuel and spare parts from the United States, there was no way they could lose.
In the end the United States forced Israel to accept a settlement (in their favor) rather than fight a larger war in the Middle East, which would have involved Saudi Arabia. However, there is no doubt that the US would have been prepared to go to any length to protect Israel, even if that meant going to war with the rest of the Middle East.