Yeah, I don't see Nasser getting couped. Basically every book on the crisis and Nasser indicated that he had an iron grip over the military, and his reforms were generally regarded as popular both by the ruling class and the general population (although these are educated guesses given that true popular support is impossible to measure in a dictatorship). The obvious parallel here is Saddam Hussein, who still retained power after getting utterly crushed in the Gulf War and even losing control of Iraqi airspace, and Nasser himself also survived a similar humiliation in the aftermath of the Six Day War.
Moreover, the Anglo-French-Israeli plan to break the morale of the Egyptians was basically strategic bombing, a strategy that has repeatedly backfired in multiple wars (in terms of harming morale at least) and only galvanized a population towards total war, which was something that Nasser was actually preparing for with his "people's war". I believe the limited bombings of Cairo IOTL already resulted in pro-Nasser demonstrations, let alone what would happen if the conflict continue and escalated. The land-based part of Anglo-French counterinsurgency tactics, based on Malaya and Algeria, were basically a "burn them all" approach involving concentration camps and razing settlements, another move that would only radicalize the Egyptians, not to mention the repercussions internationally as the war drags on longer and more media of dead civilians hit the streets of basically all Cold-War factions.
The Soviets, while folding on their nuclear bluff, would undoubtedly send covert aid to Egypt, augmenting Egypt's capabilities and/or forcing a significant escalation with a total blockade of Egypt.
Given all this, it's likely that if the US sits out, the war degenerates into a Vietnam-war equivalent that alienates the entire Third World and becomes a massive resource sink, or the US eventually gives into public and Soviet pressure and sends an ultimatum like OTL, resulting in an OTL outcome, but with higher casualties and costs.
