Btw, how did the US plan their bomber fleets to survive Soviet air defences? Did they have any cruise missiles or such paving the way at the time?
The same way they did in WW2, basically: speed, altitude, and ECM. In 1956, it had a pretty good chance of working. There'd be losses, but that was expected.
In 1956 the US was a country that hadnt had any kind of of non domestic violence within its borders for nearly 100 years so the idea that as long as it was "just"a few cities destroyed there would be no impact otherwise seems wishful thinking .They had icewater in their veins compared to most today but still. Also the idea that Western Europeans just shrug their shoulders and say -oh well that wasnt much worse than WW2 also seems totally American wishful thinking.
The USSR had 426 nuclear weapons by the end1956, delivered almost exclusively by strategic bombers (there were 24 R-5 medium-range missiles by the end of 1956, and
that was it)- which the Western powers had a pretty good chance of destroying on the ground, and a pretty good chance of intercepting any survivors. Civil defence was probably around its peak, and thermonuclear weapons weren't widely deployed.
There's a very good chance that the US doesn't get hit at all, if it's able to destroy the handful of bombers on the ground. This was a real possibility, US intelligence capability was likely sufficient to detect advanced warnings of a Soviet strike and pre-empt it. If it does get hit, it'll be limited numbers - the entire Soviet interncontinental bomber force at that time was no more than 19 Tu-95 and 34 M-4, less any that aren't launched or are destroyed in flight - and they'll have wiped out the USSR in exchange. They'll think they've won.
Western Europe will definitely be hurting. But, again, it'll be atomic, not thermonuclear, and civil defence is at its height. The comparison will definitely be to WW2 - it'll be worse, but only as a matter of scale. Again, 426 weapons - less those allocated to the intercontinental bombers - of which a substantial number are likely to be unavailable on The Day for use or intercepted before delivery. Moreover, they'll have the satisfaction of seeing that the USSR has been wiped out for it. They'll need to rebuild, but they'll see the USSR as having been the aggressor and the USA as having defeated them. They may not be happy about it, but they'll see it that way anyway.
Unless the US plan is to nuke itself to total victory what next ?
If they've just seen their plan to nuke global Communism into non-existence work? Yeah, that probably is their plan for dealing with problems in the future. You will not want to anger this version of the USA. Is it dystopian? Yeah, probably.
Anyways, I agree that it's going to depend on how this is going to start. I suppose these two options would be one of the most likely ones:
1) US tries to press the issue in Hungary using nuclear threat. Then this either escalates out of hand, or Soviets decide to call the American bluff, only for it to not turn out to be a bluff.
2) Khrushchev botches up response to Suez crisis, making it escalate to a WWIII.
You can plausibly have a combination of both, since the crises were simultaneous. It's entirely possible that the US in this timeline will blame France, Israel and the UK for causing the war, attributing it to their independent actions against Egypt destabilising relations between the US and USSR. In this case, and noting the dystopian aspects, it's quite possible that a comparatively unscathed US uses redevelopment aid to Western Europe as the 'carrot' part of a carrot-and-stick approach to bring them solidly into the US orbit. De Gaulle will be seen as dangerous for advocating French freedom of action, since French freedom of action was, in part, responsible for the war. US support to Israel may never reach the levels of OTL - and without Soviet-backed Arab forces, it may not need to. The UK more-or-less willingly entered the US orbit after 1956 anyway, but expect to see this exaggerated. Decolonisation is likely to be pursued even more aggressively - not only will the US make it a condition of aid, but the UK and France probably can't afford to fight colonial wars!
The good news is, there probably won't be the fiasco in Vietnam we all know and love. There'll still be a war in Vietnam, but without China and the USSR the dynamics will be very different. This might be where the US discovers that chucking nukes at problems doesn't make them go away. But the US might equally decide that, without global Communism as a threat, they aren't really that bothered one way or the other.