It is barely noticeable.
There is a real world example where this happend to earth (but it was estimated to be about 5000-8000 lightyears away). The "Crab Nebula" was formed by a supernova which was detectable on earth around 1050 CE. Chinese astronomers observed and described the phenomena.
It was mentioned that a bright star was visible even during daylight.
That's about it. Disappointing ;)
Earth has not only an atmosphere to protect itself from all sorts of radiation. It also has a strong magnetic field. This forms the Van-Allen-Radiation-Belt, which deflects most of the solar wind around earth.
Furthermore, our sun also has a magnetic field, which counteracts any ionized radiation/particles incoming from outside. This is the Heliosphere.
However, the effects of the magnetic fields would not help against gamma ray bursts or other purely electro-magnetic radiation, as @MikeScott and @JoshuaZ pointed out in the comments!
Gamma-radiation can even be produced on earth inside thunderstorms, those "Gamma-Ray-Flashes" have a duration (approximating here) as long as light needs to travel the wavelength of the gamma ray in question. Recordings on earth show lengths of 0.2 to 3 milliseconds. There are about 500 per day. Another source of Gamma radiation is an atomic explosion. I would argue, the effects of such radiation is highly exaggerated in media.
However, there are sensors - namely our radio telescopes and satellites - that are very sensitive to gamma ray wavelengths (or could be affected more than anybody on earth). It was observed that there is a constant flashing of gamma radiation all over the universe. These are called "Gamma-Ray-Bursts (GRB)". It is an ongoing debate where they originate from. But some star going supernova is a good candidate.
Duration of GRBs vary from milliseconds to several hours. It is said there are a few per galaxy per million years, but given the abundance of stars, they can be observed often. The energy is thousand times or more the amount of a flash. It is hypothesized, though considered unlikely, that a GRB, directly pointed at earth, could have had caused a mass extinction event (see Late Ordovician; it mentions a distance of about 6000ly. This would be not much farther away than the Crab Nebula. The cause is supposed to be a "Hyper-Nova" though, not a regular Supernova. Seems far fetched to me).
On a general note: Remember that a supernova is a star dying. That means it is like our sun, the center of some other solar-system. The closest solar-system we know of is about 4 lightyears away (Proxima Centauri). And distances get only longer after that. To really be affected by a supernova, you would have to be in the same system as that star, probably. This might or might not turn out as in the cases you described. It surely would be devastating, if survivable at all.
But it is true, that such radiation would hit us without possibility of detecting it prior to impact. It is because we can only look "into the past" if we watch any star. The light that is on its way is not detectable, because it needs to hit a sensor (or eye) first.
Grave thought, if our sun would go supernova: Since the sun is about 8 lightminutes away, and we have satellites orbiting it, if ground control notices that it had lost contact with the satellites, the inbound supernova/radiation travelling at lightspeed is imminent to impact at the same time. No time to kiss anyone goodbye - That is if the sun would explode from one second to the next.
It is very probable that a star will send signs of a changing situation much earlier and less violently. For example by increased solar winds over a long time, or the circumference expanding a bit, then contracting again or something like that (see Betelgeuse of the Orion constellation, a red giant supposed to go supernova "sometime").
After such events it might still be millions of years until it will explode. Our understanding of the sun is not yet advanced enough to know for which signals to look in order to predict anything, though. We just started observing our sun in a level of detail that might lead to some insights one day (see SOHO satellite for example).