Very interesting question!
In chemistry you spend lots of time discussing exothermic and endothermic reactions: when you put your reagents together, sometimes the reaction heats things up, and sometimes the reaction cools things down. Nuclear reactions are very different, in that essentially all spontaneous reactions studied in laboratories are exothermic.
However, there's an important difference between a nuclear laboratory and a chemical laboratory: temperature. In a chemical reaction, the energy scales involved in making and breaking bonds may be a few electron volts. For example, water's latent heat of fusion, 330 J/g, works out to about 60 milli-eV per molecule. A room-temperature heat bath has lots of phonons with mean energies around $kT = 25\,\mathrm{meV}$, so finding a 60+ meV phonon to break a water-water bond is not unlikely.
By contrast, typical nuclear excitation energies are millions of eV. Endothermic nuclear reactions don't happen spontaneously for the most part because laboratories here on earth operate at zero temperature, as far as the nucleus can tell. There just isn't any heat to suck up to drive the reaction.
In a hot environment, such as the core of a star, the story is different. There you start to have enough energy that you can trigger exothermic reactions with energy barriers, such hydrogen-to-helium fusion. But even in that case you don't have much contribution from endothermic reactions. (If there were lots of endothermic nuclear reactions happening in the cores of stars, they'd suck the heat out of the core until it was too cool to drive the reaction.) There are a few counterexamples. For example, deuterium disintegrates if it absorbs a high-energy photon; but since deuterium is an intermediate stage in proton-proton fusion, the effect of this is that the deuterium acts like an unstable nucleus whose lifetime depends on the temperature.
For another example, consider the production of uranium from lead. This is clearly an endothermic process, since uranium, left to its own devices, will decay into lead, an ensemble of alpha and beta particles, and heat. Uranium is produced from lead by a series of neutron captures, which release energy, and beta decays, which release energy; the free neutrons come from fusion reactions between alpha particles and moderate-mass nuclei, which release energy.
All of the steps from lead to uranium are exothermic.
How is it that an endothermic reaction can be made up of exothermic steps?
There are a couple of things to consider there.
First is that, since the reaction does not occur at thermal equilibrium, there are several processes competing whose timescales must be taken into consideration.
Here's a diagram showing the lifetimes of the nuclei involved; isotopes in black are stable, while lighter colors have briefer lifetimes:
If the neutron flux is low, element production stops at lead and bismuth: neutron capture on bismuth-209 gives bismuth-210, which beta-decays after a week to polonium-210, which alpha-decays after a few months to lead-206.
It's fair to think of the cycle where lead-206 absorbs four neutrons and emits two betas and an alpha as a sort of catalyzed, exothermic fusion reaction.
However if the neutron flux is high, the unstable isotopes may absorb neutrons themselves, which is the route to the longer-lived isotopes around uranium, thorium, and radium. In some sense, when we take the radioactive heat from a block of uranium and say "this energy was stored here by an endothermic process in a long-dead star," what we've actually done is to interrupt the conversion of neutrons to alphas via lead at a particularly long-lived intermediate point.
The other important distinction between endothermic uranium production and endothermic reactions in chemistry is that the nuclear reactions are exchanging both heat and particles with their environment; it takes much more care to distinguish between "the system" that we're interested in and "its environment" which provides the heat.
This is a stark contrast from a chemical reaction where only heat flows between the system and its environment, and the reaction is driven by its entropy.