I agree with you on the steam cycle and the process of heating and cooling the water, but I disagree with your statement that:
“It makes no difference whether this steam was created by nuclear fission, or burning fossil fuels.”

As far as the water molecules are concerned, I'm sure they don't care what kicks them into higher velocity, but there is a huge difference in the way the energy sources work.

Imagine that we go on a unusual camping trip, our aim is to boil potatoes at a camp site.
Both of us have a similar containers and we get water from the well at the camp.

You take with you some coal and a means to light the coal. The coal needs to be heated before it starts burning, it may take some lighter fluid and a match to ignite or start the chemical reaction within the coal, when the chemical reaction from the heated coal becomes energetic enough, then you place the container of water on top of the burning coal and you can drop the potatoes into the water and soon you will have boiled potatoes.
After enjoying some boiled potatoes you wait for a while and the coal fuel runs out and ceases to give off any energy.

I on the other hand want to be different, I take a quantity of uranium metal with me, I also need to ignite the energetic process within the material; in nuclear physics the ignition is called chain reaction, It can be done several different ways, one method is to simply pile enough of the uranium in one place so that the natural fission reactions within the uranium will increase the number of atoms that are split in the pile. I'll use this simplest method even though it requires a lot uranium. There is the problem though that this reaction could go out of control, producing more energy than needed, in that case the potatoes and most everything in the camp grounds would be fried instead of boiled. It is lucky that there are other materials that can can be used to slow down the reaction.

After I get the desired reaction going, I can put the container of water on the pile and boil my potatoes. There might be some concerns though, the potatoes might be hotter than your potatoes because the neutrons and gamma rays may have penetrated into the potatoes and given the molecules in the potatoes some extra motion, perhaps even making them radioactive. Another problem is that I will not be able to quench this fire, it will continue for a very long time, of course our great great grand children could boil their potatoes there also.

The question that arises is: What happens to the energy that heats the water in our mental experiment?

In your experiment the chemical reaction causes the surrounding molecules to move faster, in other words, generate heat, once the reaction stops the molecular motion returns to the level where it was before the reaction was started.

In my experiment, once I start the reaction I can not stop it, The reaction will continue at an accelerating pace, unless the material is distributed so that there is more space and also cooling between the segments of the energy producing material. Such action will only stop the accelerating pace but not the energy released in the form of radioactivity that causes the heating of the surroundings.
The induced splitting of the atomic nuclei will continue for a very long time, see the link below.

The oldest working reactor is in Russia, It was started on December 25, 1946:
https://en.wikipedia.org/wiki/F-1_(nuclear_reactor)

http://www.ccnr.org/decay_U238.html

From the above site: “Depleted uranium remains radioactive for literally billions of years, and over these long periods of time it will continue to produce all of its radioactive decay products; thus depleted uranium actually becomes more radioactive as the centuries and millennia go by because these decay products accumulate.”

Radioactivity is the energy that heats the water in a nuclear reactor. Uranium is more energetic when it is taken out of a nuclear reactor than it was when it was placed in there.