I don't know about the non-Russian republics, but in Russia, after the fall of the USSR a new ministry was created to manage the nuclear power plants (Minatom), which took over all of the Soviet responsibilities. This was more or less still on the "Soviet template" as I understand it. In 2007, however, a new law was passed that got rid of Minatom and replaced it with a new, not-for-profit "state corporation," Rosatom, which took over these responsibilities. So it is a kind of strange entity, being both the state, and a corporation, and a not-for-profit — it doesn't quite fit into any single category easily, which is a nice metaphor for a lot of the corporate-state hybrids that emerged in the years after fall of the USSR and esp. the rise of Vladimir Putin.

Beyond that — I don't know the answer. There is a lot of historical literature on the development of the Soviet nuclear complex (which went through complex bureaucratic phases as it started as a wholly military entity, then transformed into a civilian-military hybrid system), but I haven't seen much written on their post-Soviet changes (it would be a good research paper topic, the sort of thing I might assign a student to work on in the future!). The questions about employee turnover, training, etc., are great ones, since the Soviet model for employment was pretty different than the post-Soviet one (for the Soviet nuclear industry, you were basically "assigned" to it, you didn't necessarily choose it yourself — I doubt that continued in the post-Soviet era).

For the details on the development of the Soviet nuclear industry, Sonja Schmid's Producing Power gives a very detailed account up the entire Soviet period (mostly pre-Chernobyl, but she covers some of the changes that came afterwards as well, but not much of the post-Soviet period, I don't think).