I was watching Solaris (1972), in which the main character says: "Man is the one who renders science moral or immoral. Remember Hiroshima." While I am curious about the line would have meant in the context of the film, I am more interested in the immediate response of the Soviet state and people.
In terms of the immediate reaction, it was barely reported on. Hiroshima got scant news coverage and Nagasaki got essentially none. This omission was deliberate. Under Stalin, nuclear weapons were basically not discussed in the Soviet press. Of course, the true "immediate response," by Stalin, was to authorize the creation of their own nuclear weapons as well, but that was secret, and it was not really until after Stalin's death that the Soviets acknowledged having nuclear weapons.
That doesn't mean you couldn't know about them or learn about them with some effort — the Smyth Report was translated into Russian and widely available — but it was not a topic of endless discussion like it was in the West. Why not? Because Stalin interpreted the Western emphasis on the atomic bomb as indicating an advantage over the Soviets and didn't want that part of the national discourse. So the official approach was to not talk about them, and to dismiss them as irrelevant. One saw similar things in China under Mao, until they had their own weapons; you downplay the threat by pretending to ignore it.
Of course, just because your newspapers don't talk about something doesn't mean people don't know about it and don't think about it. But the Soviet emphasis in World War II was always their own involvement (esp. with the defeat of Japan), and that even continues today in post-Soviet nations. So there is not a lot of indication of a distinct "response" by the Soviet people; it was entirely downplayed by the state.
It was not really until Khrushchev that nuclear weapons, and Hiroshima/Nagasaki, became part of the national discussion. The basic framing was always that the US built these weapons first, and was the only nation to use nuclear weapons against civilians in war, and that the Soviet nuclear arsenal was defensive in nature against this threat, and so on.
By the time of the early 1970s, however, there were more complex discussions and critiques of the development of nuclear weapons, some of which were led by Soviet scientists themselves, like Andrei Sakharov. So one should not take the 1970s as being indicative of the attitude in the 1940s.