To borrow a phrase from XKCD: "You wouldn't die of anything; you'd just stop being biology and start being physics."

Obviously we experience the results of nuclear events any time we're out in the sunlight. However, we receive those results via electromagnetism--and gravity, because the sun affects the tides. Nuclear forces govern the fission reactions at the heart of nuclear power, but the heat is transferred via EM forces. Even in the event of a nuclear weapon explosion, the gamma ray pulse is EM, the thermal pulse is EM, the visible light is EM, and when the blast wave hits, it's doing damage by EM forces, too.

Maybe it's just my lack of imagination, but I don't see how a person could directly experience the strong or weak nuclear forces without being part of a significant fission, fusion, or decay event.

However: it turns out that only about 1% of the mass of a proton is composed of the rest mass of the quarks that make it up. The other 99% is the binding energy holding everything together, which is an effect of the strong nuclear force.

So, get a liter bottle of water and wave it around. Feel the heft. You're pushing on that bottle via electromagnetic forces, but 990 grams of that mass you're playing with is nuclear force.

Acute radiation sickness. Specifically from neutron radiation. Even that is in a sense indirect - the neutrons knock nuclei about or get captured by the nuclear forces, but it's secondary emission of ionising radiation that does the real damage and that's electromagnetically.

You have likely experienced them directly when a cosmic rays hits you, otherwise you experience them indirectly from sunlight.