July 18: Hungary's Stalinist premier, Rakosi, removed after Khrushchev denounces Stalin.
July 19: The U.S. withdraws financial support for the Aswan High Dam project.
July 26: President Nasser of Egypt nationalizes the Suez Canal.
August 1: The U.S. ambassador, meeting with the French and British foreign ministers, insists on diplomatic resolution of the Suez crisis.
Oct 16: Hungarian student groups submit demands to reform Communist rule.
Oct 22-24: At the Sevres Conference France, Britain, and Israel secretly agree to jointly invade Egypt; France agrees to transfer nuclear technology to Israel.
Oct 23-25: Mass protests against Communist rule begin in Budapest; Premier Gero requests Soviet intervention. Hungarian government falls, Imre Nagy becomes Premier; facing massive resistance, Soviet troops pull back to the countryside to regroup.
Oct 29: Israeli forces invade the Sinai Peninsula.
Oct 30: At a meeting of the Soviet Presidium, Marshal Zhukov calls for Soviet withdrawal from Hungary and recognition of the Nagy government. The next day, Khrushchev reverses the Presidium's verdict, and orders the invasion of Hungary.
Oct 31: Britain and France begin air operations against Egyptian targets; Egypt sinks ships to block the Suez Canal.
Nov 1: Soviet forces invade Hungary. Hungary declares itself a neutral state. Four days later, Budapest has fallen to the Red Army.
Nov 5-6: British and French airborne and amphibious landings seize Port Said, Egypt.
Without Suez to distract them, Ike and Dulles might have been able to neutralize Hungary. (Condemning the invasion of unruly client states apparently cut both ways in October of 1956, if at no time before or since.) Without Suez to bolster the hardline narrative, Zhukov's sensible counsel might have won the day permanently. Result? A slow-motion disintegration of the Soviet Empire, as more and more nations opt for a Titoist or Nagyist transition through neutrality, democratic socialism, and then ... what? Without an expensive empire to run, do the Soviets weather their 1980s fiscal crisis successfully? Without the specter of Soviet oppression, does NATO fall apart in Europe? Or is Khrushchev succeeded by another reformer, pulling the whole Soviet Union in Hungary's wake?
Without Hungary stampeding the Americans into putting their foot down, Eden could bring Nasser to heel and perhaps stop (or at least slow) the disintegration of the British (and French) colonial empires. At the very least, by preserving the credible threat of independent action, Britain and France could have remained legitimately Great Powers for the rest of the century. Maybe the African states stay in closer association to the metropoles; the fall of Nasser (inevitable after a loss of face at Suez) alters the whole nature of Arab nationalism and reinforces the prestige of the European-supported monarchies.
What if the Sevres Conference pushed the invasion timetable down the road until spring of 1957, counting on economic warfare to weaken Nasser domestically in the interim? If you're a fan of multipolarity, maybe the "late Suez" is your uchronia: Hungary neutral (with a defense pact with Yugoslavia, perhaps) and Egypt's ears pinned back as the newly confident European powers start looking around for more crises to solve with a regiment of paras and a couple of aircraft carriers.
Lots of little possibilities there, which Eisenhower's anti-imperialism quashed when he threatened to starve and bankrupt Britain rather than let them keep Suez. (It really is a special relationship, all right.) Could a less anti-imperialist (or more Europhile, which is not necessarily the same thing) President have swallowed the contradiction and supported both Suez and Imre Nagy? Perhaps a weaker President, less sure of his foreign-relations strength (Stevenson?), might have let England and France take the lead and sold out Hungary even rhetorically; perhaps a more isolationist President (Taft?) would have let both Nasser and Nagy be crushed.
Or, the whole thing could have gone pear-shaped as World War Three starts at two simultaneous locations. It would have been a WWIII without ICBMs, which means the United States would have come through a conflict that devastated Europe almost entirely unscathed a third time. And Europe would surely have been devastated, even if "Operation Dropshot" worked perfectly and beheaded the entire Soviet military-industrial complex in mid-November 1956. (On balance, mostly depending on whether you think Mao would have fallen from power as a result of the Soviets' destruction, net lives might even have been saved over OTL.) But as the smoke clears, the British are secure in Egypt and the French are secure in Algeria, Hungary leads a community of free nations in Eastern Europe, and Eisenhower launches a new Marshall Plan to rebuild a chaotic (most likely military-governed) Russia.