Since there are many upvotes here, and people are going to perhaps wonder why there aren't good responses, let me say: this question gets asked a lot, and I never answer it anymore. Why? Because a) "other nations" is too broad a category (name the ones you care about — no answer on here is going to generalize well for +100 places), b) "other nations" doesn't really specify who in the nations you might care about (the general population? the political leaders? the military leaders? each of these will have different reactions), and c) even if you are very specific, it is quite difficult to give good answers for more than a handful of cases anyway, because widespread public opinion polling barely existed even for the United States (there are a couple polls from 1945, but that's it, and their scientific rigor is pretty dubious), and there just isn't a ton of scholarship about how every nation reacted to the atomic bomb anyway. It is one of those historical questions that seems well-formed on the surface but is in fact very, very vague, which becomes clear if you try to think about answering it in a rigorous way.

We could look at subsets of this question quite fine:

• How did Soviet leadership react to the atomic bombing of Hiroshima? Stalin sulked in his dacha for a day, because he feared that the Japanese would immediately surrender, but when they didn't, he accelerated their plan to declare war against Japan and also accelerated his own atomic weapons program. (We can't really answer this question for "the Soviet people" — aside from having essentially no data to base the answer on, the Soviet people were largely kept in the dark about the events at Hiroshima and Nagasaki, because the Soviet state officially wanted to downplay the atomic bomb until the USSR had their own; it is a reasonable generalization that during the Stalin period, there was very little public discussion of nuclear weapons in any form, it was not until the Khrushchev period that this became a staple of Soviet public discussions. Hiroshima was given very limited press coverage in the USSR, and Nagasaki was basically ignored by Soviet state press, as an aside. This was noted with great interest by the United States!)

• How did the peoples in states subjugated to Japanese rule, like China and Korea, think about the atomic bombings? Very broadly speaking, they were pretty favorable towards them. No big surprises there, but it is somewhat interesting to note, and various studies of, say, Korean responses to nuclear technology in general usually start with this fact.

• How did the British leadership think about the atomic bombings? The United Kingdom was part of the group that made the atomic bombs, so it wasn't a big surprise or anything; they were required, under the Quebec Agreement between Roosevelt and Churchill, to assent to the bombings, which they did readily.

• Interestingly to fans of the Civ game series, Gandhi downplayed the significance of the atomic bombs. Gandhi's views should not be construed as representing all of India, obviously.

But breaking it down much further than that becomes quite difficult, or relies on extremely broad generalizations that are hard to support. Even for the US is it hard to answer the "who" question. All Americans? All Americans of a certain age? Of a certain race? Of a certain political party? Of a certain amount of wealth or power? We just don't have the data for that. Modern polls on attitudes towards nuclear weapons break down the demographics a lot, and as you'd expect they vary a lot. We would expect the same to be true in 1945, and what evidence we do have does indicate this (e.g., the editorials in African-American newspapers had a different take than those of national US newspapers), but we don't really have great ways of talking about it in more than generalities.

The only countries where to my knowledge there have been really systematic studies of their reactions the atomic bombs, by looking at a variety of different kinds of "data" (survey polls, cultural products, editorials and articles, etc.) are the United States and Japan. In the US case, Spencer Weart's Nuclear Fear is great, as is Boyer's By the Bomb's Early Light. It would be great to do studies like this of other nations, but I haven't seen the equivalent sorts of in-depth studies. It should be noted that in the case of the US (and Japan, but I am less familiar with the depth of that literature), there are many reactions to the atomic bombings. There was not a single "national" reaction in any meaningful sense. Even amongst politicians, military generals, etc., the bombings were controversial, and were a mixture of feelings of excitement, enthusiasm, and fear for the future. One would expect that other nations were similarly complex in their spectrum of possible reactions.