Will Czechoslovakia being a member of the "Western Bloc" change the outcome of the Hungarian Uprising in 1956?
If we assume that all of Czechoslovakia is actively pro-Western and not merely neutralized, that means Hungary shares a border with NATO. Also, that East Germany, which would assuredly be tightly under Soviet control has a much longer border with the West, and if Austria is more tightly under Soviet rule, it too is a long salient mostly surrounded by pro-Western territories.
All of which points to a much higher degree of Soviet paranoia about holding these countries, but how can the Russians keep control most effectively?
OTL the Soviet hegemony was rarely a simple matter of conquest; almost everywhere they had local allies. Even Poland, which is the most obvious exception, had its people who would freely align with the Soviets, it's just that there they'd be very much in the minority.
Austria though might be an extreme case comparable to Poland and it seems likely to me that ITTL the Soviets would mostly let it go, once satisfied they'd be neutralized in the sense of agreeing not to join a pro-Western, anti-Soviet pact like NATO (or become the kind of "neutral" Sweden was, genuinely detached from the operational alliance but de facto a well-armed pro-Western ally).
But with Hungary directly bordering a NATO nation, what will the Russians do? We could look at Bulgaria and of course East Germany for examples.
Arguably they'd take more care to build consensus for the Warsaw Pact in Hungary and thus avoid the crisis in the first place, but that's difficult. There are people in Hungary who would never be content with the Russian alliance, and people who would never accept a socialist economy; to try to buy off others who are wavering would involve more success in the economic front and that was something I presume the Soviets did their level best to succeed at, so they can't really offer more in reality. The alternative is a heavier-handed crackdown from the beginning, which of course would merely alienate more Hungarians sooner and give legitimacy to any revolutionary movement whatsoever, even one that is obviously reactionary.
And if the internal crisis does develop as OTL, Eisenhower or whoever might be US President come the mid-1950s would not have the excuses he did OTL, that Hungary was simply inaccessible to US power. Of course anyone sane looking at a map would see that using NATO force to intervene inside Hungary would be asking for a very nasty war, what with Czechoslovkia--which would be involved up to the eyebrows, being the only Western conduit to Hungarian borders--being so terribly vulnerable on at least one huge flank to the north, on two that have the place literally within Soviet jaws if Austria is being held as a full WP/COMECON member as well.
So Ike might not be able to say what he said OTL, but the Czech government might be the one to veto such intervention, pointing out their country would surely die. But then again, if there were an all-out war started by either side, the Czechs are right there to be blasted and chewed up in short order anyway, so if they were to say that in this case, there's some doubt they'd rally to the war in any case. Presumably they might be resolved to join the fight in a less ambiguous war. Then again maybe they'd be gung-ho enough to have the fight over with that they'd actually pull NATO into the war against its better judgement at that point.
Looking at the very exposed position Czechoslovakia is in, I think it might be possible they would not be members of NATO at all, but neutral as Austria was OTL.
In that case, the West's options regarding the same sort of crisis in Hungary at the same time would be no different than OTL. Therefore the Soviet options leading up to it would be the same and we could expect Hungary to go much as OTL.
). In the Suez Crisis of 1956, Israel was allied with Britain and France against Egypt, but Eisenhower pulled the plug on the whole adventure when he threatened sanctions in the form of stopping oil subsidies to both the European powers and demanded they withdraw and settle the crisis through the UN.
