I think this boils down to a question of how much fundamental understanding is needed to build a very basic nuclear reactor to boil water.
If fissile material was readily available to an otherwise primitive steam-age civilization, could they use it as a fuel source without having a deeper understanding of atomic physics? Is simply understanding that when you place these special rocks in close proximity they get hot, enough to make a steam locomotive?
I think it matters how the fissile material is "lying around". In our world, the two limiting factors are the mining and extraction of uranium, and the enrichment of the fissile isotope. Of these, the second is substantially harder (requiring gas centrifuges and other fairly 20th century technology) - the former is feasible with only late 18th century refining - pure uranium salt was isolated in 1789.
As pointed out in the comments, enrichment is not actually necessary for modern reactors - the Chicago pile 1 ran on natural uranium. However, it seems unlikely that anyone would realise the potential to generate atomic energy without a firm theoretical motivation, which would not be possible without an understanding of atomic science. I postulate that the only way to see fission happen without understanding it fundamentally is by having your natural uranium be "highly enriched" by our standards.
Suppose that your uranium occurs naturally in a highly enriched isotope mix. I see the timeline going as follows:
The Manhattan Project was limited by three things - rarity of uranium ore, poor understanding of the fundamental physics, and the difficulty of enriching the uranium. Much of the scientific work that went on at Los Alamos was in fact pure trial and error - empirical measurements of cross sections and critical masses, etc. If your uranium is both naturally enriched and abundant, you short-circuit that. Aside from this, early atomic piles did not strictly speaking require any technology that could not (in principle) have been made in the late 1800s.
In short, I believe that an abundance of dilute, enriched uranium could push reactors forward 50 years, perhaps more, but an advanced industrial mining complex is still needed to extract and purify useful quantities of raw material. Reactors can be extremely primitive indeed - one even formed by accident. On a human scale, a sufficiently large block of U-235 with a control rod will function as a reactor - but the process of getting to a safe design will be, erm, messy.
If we assume an Earth-like world, where life evolved about three times as fast, then they need no technical understanding at all.
One of the reactor zones at Oklo, Gabon, Africa (image credit: US DoE)
When the Earth came to be, the isotope composition of natural Uranium was — roughly — the same as that of present day nuclear fuel, after enriching.
This then means that what was needed to get a chain reaction going was...
...and this is exactly what happened at what is present day Oklo, Gabon.
So, 2 billion years ago, for tens of thousands of years, these natural reactor zones operated in cycles. Water would flow through the ore, moderate the chain reaction, cause the zone to heat up, make water to boil off, the chain reaction stops, the zone cools down, and start over, in about 3 hour cycles.
Assuming a primitive culture existed back then, they would be able to see this happen with their own eyes.
The issue would be, that this would not be very healthy for them.
But, yes, they would be able to see that if you pile that very particular kind of dense and heavy rock together in large enough a pile, and soak it in water, then it heats up.
...and then they get sick and possibly die, without knowing why.
Yes and No.
The core idea has enough merit that it could be plausible. However just getting things hot enough to boil water is the easy part.
Doing it in such a way that you don't accidentally irradiate everyone and kill them off in horrible ways is much much harder.
And therein lies the problem with this idea. With modern physics, it took us nearly 30 years to discover that radiation was actually quite bad for you and even longer still to fully appreciate that.
But let's assume that because of the amount of Fissile material lying around, our inhabitants have evolved some resistance to radiation:
Without an understanding of what is happening - the chance of having a runaway chain reaction leading to a meltdown or nuclear explosion increases drastically
Imagine you are but a simple worker:
If one lump of magic rock is good, 2 must be better. If two lumps of magic rock is better, then 3 must be amazing. If three lumps of magic rock is a Amazing then 4 must be Nuclear detonation.
Nuclear reactors are simple in principle but in practice are complex and delicate machinery, building and operating them safely requires a good understanding of the underlying processes. Given the dangers it's unlikely this understanding can be developed by pure, unassisted trial and error.
Frame challenge: Radioisotope thermal generators avoid most of these pitfalls.
While still needing extremely dangerous materials (let's not forget that fissile materials are typically also highly toxic), it is just about possible for an early civilisation that can already extract, smelt and process lead to discover that certain Deadly Rocks can be encased in lead and be used to heat stuff.
The crucial thing about RTGs is that not only are they mechanically simple (no moving parts, no special materials, just some fissile stuff surrounded by heat transferring shielding), they use fissile isotopes incapable of sustaining a chain reaction.
There is still a lot of experimentation required to do this reliably and without the heat melting the lead casing but depending on the relative abundance or scarcity of other sources of heat it could be a tempting enough option.
Especially if we consider that in a society that already has much higher mortality levels than rich modern societies, detecting longer-term harms of radiation would've been much more difficult. Just look at how long it took even as recently as the 19th century to establish that stuff like arsenic-laced wallpaper, gas lighting or coal-burning fireplaces were slowly killing you and thus weren't great to have in your home.
The main challenge with RTGs however is that while they last an awfully long time, their power density is fairly low, and they need isotopes with relatively short half lives, so there must be a natural process that continuously produces lots of them. As no such process exists on Earth, real-life RTGs rely on isotopes created in nuclear reactors, which puts us back to square one: you need to build a nuclear reactor first.
Maybe and no.
Nuclear reactor as such is not an issue at all: they can occur spontaneously, too, as already pointed out. The issue is basically the same as with the steam in steam engines: producing steam is not a problem at all: it can occur naturally as well.
The actual problems lie in everything else but the power source.
You can revisit the question by asking if the civilization could create the same thing by burning wood, ie. obtaining the heat from combustion instead of nuclear fission.
If the answer is "no", then they cannot build the same thing with nuclear power source either. Which is probably the case.
However, if the answer is somehow, miraculously "yes", then it is somewhat plausible that they could end up using fissile nuclear material instead of combustible one, but there are still some extra issues left: eg. shielding against radiation, controlling the chain reaction (it needs way more precise control than a furnace), and handling the waste.
In my opinion, there are too many issues with the trial-and-error approach that I think it is almost implausible, but not completely impossible. Science, after all, has involved lot of trial and error, too, but they also tried to understand what was going on, and make hypotheses and test them etc. It is hard to believe that all of this could be achieved just by trial and error, ie. without something resembling science.
For a crude nuclear reactor, you basically need three things: A fissile fuel (Uranium-235), typically in a concentration of 3%-5%, a moderator (water) to slow neutrons down to thermal energies (this increases the probability of fission due to reasons), a coolant to transfer heat away from the reactor (also water). You might also want a strong neutron absorber like boron to control the reactor, but this can also be achieved by removing fissile material.
If you assume that your world conveniently already has some naturally enriched uranium, then it's not that hard to create a reactor, and I could see it being discovered by accident. Like someone was storing a bunch of uranium ore, and it flooded (adding a moderator), resulting in a chain reaction. From there they might just start putting uranium into a big water tank and seeing what happens. In this scenario, it's actually kind of hard to have a meltdown, since boiling water will displace the moderator and slow down the chain reaction. This is known as a boiling water reactor (BWR) if you want something to google. Now it's not impossible to have an accident with a BWR.
Keep in mind though, when nuclear power gets discovered, the conversation usually goes like so: "Hey there's way more energy stored in them there glowy rocks than any of our fossil fuels", "Yeah, I bet we could make a pretty kick-ass bomb with that stuff".
No, a primitive civilization would not be able to trial-and-error their way to a functional nuclear reactor without a deeper understanding of atomic physics. Building a basic nuclear reactor requires not only access to fissile material but also a fundamental understanding of criticality, radiation, heat transfer, and reactor dynamics. Without this knowledge, it would be extremely difficult to harness nuclear energy safely and efficiently.
