Greetings! A most interesting question here, and one which is actually debated to some extent within Cold War academia, or at the very least featured alongside the impact of the Suez Crisis and the Polish October (both of which occurred around the same time as the Hungarian Uprising). This response will deal more with the geopolitical impact of the USSR's violent crackdown during the Hungarian Revolution, rather than the popular memory of the 1956 Uprising and its immediate treatment in the following years.
Note: Due to OP specifically asking about the "reaction" of superpowers towards the Hungarian Uprising, this response will not go too far in-depth about the causes, and course of the Revolution in favour of focussing on the consequences. Further reading on the origins of the events which gripped Hungary are included in the sources.
A Difficult Position (USA)
The Hungarian Uprising occurred during a backdrop of other developments in the Cold War. Between October and November of 1956, the superpowers of the new world order had had to lambast the superpowers of the old world order in the Suez Crisis, when Britain and France (with Israeli support) militarily intervened in Egypt following the nationalisation of the canal by President Gamal Abdel Nasser. Whilst Khrushchev "rattled his rockets" and threatened nuclear strikes in Paris, London, and Jerusalem if the invasion was not called off. US President Eisenhower, for his part, also protested the Anglo-French-Israeli invasion, furious that they had not sought prior approval with America. He however, threatened the European allies with economic sanctions rather than full-blown nuclear retaliation, and did so quietly rather than publicly. Because of this, the public (at the time) was unaware of the American government's involvement in the protesting of the Suez Crisis, but they were aware of the Soviet government's brash threats.
This complicated matters significantly for the USA and other Western European governments. The Red Army's violent invasion of Budapest against the Hungarian "rebels" coincided with the Anglo-French landings near the Suez canal. The former occurred on November 4th, and the latter on November 5th. If Washington chose to lambast the Soviets for brutally crushing the Hungarian Uprising before stepping in to deal with the escalation (by their Western European allies no less) in Egypt, then their prestige in the eyes of the world would decrease significantly. Richard Nixon, who was Vice-President at the time of Suez and Hungary, outlined this conundrum rather well:
"We couldn't on one hand, complain about the Soviets intervening in Hungary and, on the other hand, approve of the British and the French picking that particular time to intervene against [Gamal Abdel] Nasser"
President Eisenhower also bemoaned this "colonial side-venture" which seemed like an anachronistic event in the postwar order of the 20th century.
"How could we possibly support Britain and France, if in doing so we lose the whole Arab world?"
Linking back to the Hungarian Revolution here, the poor timing of the Anglo-French-Israeli invasion in the Suez Crisis, which caused America's hands to be tied in Africa rather than Europe, meant that they could little in Hungary (short of propaganda) to influence the turn of events there. Further, the western powers had accepted by this time in the Cold War that Hungary was firmly in the Soviet sphere of influence. This Revolution was not one happening at the fringes of the Soviet "monolithic empire", where the West might be able to leverage some influence, but it was instead occurring in a country which was geographically within Moscow's Eastern European "zone". Historian Ian Kershaw remarks along similar lines:
"The Western powers were conveniently mired in the Suez crisis - through, whatever their sympathies for the Hungarian rebels, it was clear that they had no intention of risking a possible world war through intervening in the Soviet sphere of influence."
Thus then, the reaction (or lack thereof) of the USA to the Hungarian Uprising confirmed what had been suspected since the early years of the 1950s, when Soviet control over Eastern Europe had been established. In only applying the "amorphous" pressure of propaganda and press-based condemnation, the US and Europe acknowledged the boundaries of their zones of influence. Hungary was to remain a country in the Soviet zone of the Cold War, even if that meant the use of force.
The Revolution Betrayed (USSR)
Whilst Washington was occupied over the Suez, Moscow was also dealing with the consequences of the "Polish October" reforms which had helped to ignite the Revolution in Hungary. The Kremlin was divided over the option of military intervention in Hungary, or allowing the rebels under Imre Nagy to end one-party rule from Budapest. Context is necessary here as well. Khrushchev had made his "Secret Speech" denouncing Stalinism in February of 1956, and the Communist Party in the USSR had arranged for the removal of Stalinist dictator Matyas Rakosi from power in Hungary (which occurred officially on health grounds in July of 1956). When Moscow replaced him with Erno Gero, however, this move was met with public outrage. They had been calling for Imre Nagy, whose ideas of "democratic socialism" were highly preferred to the tightly-controlled and centralised form of Communism that had plagued the country under Rakosi. Over a series of skirmishes with the Red Army troops and back-and-forth between Budapest and Moscow, Soviet troops finally left the city (or so it seemed), on the 29th of October. '
This all changed however, when it was revealed to the Kremlin by their emissaries in Budapest, Mikail Suslov and Anastas Mikoyan, that the new Hungarian leadership under Nagy were calling for a "neutral Hungary" which would no longer be a member of the Warsaw Pact. This brought with it a series of new considerations and concerns for Khrushchev and the Kremlin Politburo. They feared that Soviet withdrawal from Hungary and allowing it to leave the Warsaw Pact would not only expose a weakness to be exploited by the Western powers (which we must remember, were at that time also attacking the USSR's ally, Egypt), but that it would fuel further dissent across Eastern Europe. Ian Kershaw notes this as well:
"In favour of intervention, beyond the worsening situation in Budapest, was the worry that any sign of weakness by the Soviet Union would be exploited by the imperialist Western powers. A far more serious consideration was that Soviet weakness would encourage the spread of unrest - already being reported in neighbouring Romania and Czechoslovakia - to other parts of Eastern Europe. The risk of contagion was a grave one, as the Soviet leaders were well aware. It was the critical factor.
Thus when the Red Army launched its full scale assault on Budapest on the 4th of November, it proved brutally to the world that the Soviet jackboot was resting firmly on Eastern Europe, and that despite the perceived move away from the old Stalinist order, Moscow would tolerate dissent no more than it had during the early years of the 1950s. In Western Europe, Communist parties which had once looked to the Soviet Union as their model and guide, began rapid declines in popularity as members deserted en masse. Anne Applebaum on this shift in pro-Communist sentiments in the West:
"After 1956 the French communist party fractured, the Italian communist party broke away from Moscow and the British communist party lost two thirds of its members. Even Jean-Paul Sartre [a leading French Marxist] attacked the USSR in November 1956, though he retained a weakness for Marxism long afterwards."
Yet within the larger course of the Cold War, this brutal repression of the Hungarian Uprising merely affirmed to the world an assumption which the US had deeply rooted in its strategic and geopolitical views of the USSR: that Eastern Europe was firmly in the Soviet zone of control, and any attempts from within or without to rebel would be ruthlessly crushed. Though Hungary in 1956 was a bloody reminder of this reality, it did not contribute anything particularly novel towards the heightening of tensions or the expansion of the Cold War. That would come soon after 1956, with the emergence of Cuba as a key "battleground" for the superpowers, and with the Suez Crisis' expansion of the Cold War into Africa and the Middle East.
Part 1 of 2