Is it an extremely painful way to die? Does nobody survive?

seashellz is correct. The book describes it in some detail -- diarrhea, vomiting, "bloody stools" and so forth. The blood-forming tissues are destroyed, causing a loss of body salts, according to the novel. It also says that death occurs from exhaustion, meaning the body just dying off, or in some cases infection may be the immediate cause of death. Depending on a person's health and degree of resistance, it could take a week or so to die.

An interesting aspect in the book is that it's said you can fall ill for a couple of days and then experience a brief recovery, lasting perhaps a week to ten days, before falling ill a second and final time. This happens to the character of Peter Holmes in the book. People who are heavy drinkers also seem to have a greater resistance to radiation due to alcohol, though they of course finally succumb to its effects. I don't know if this is true but I assume the author Nevil Shute researched it pretty carefully. Also, animals have greater resistance to radiation and will outlive humans by several months, with the rabbit being the last to go, about a year after mankind.

Of course, the inhabitants of Hiroshima and Nagasaki experienced radiation sickness as a consequence of the atomic bombings (apart from the immediate effects such as burns and so forth, which would not occur in an On the Beach scenario), but I don't know how close Shute's description aligned with those experiences.

The final question is whether this scenario -- of all life being extinguished in a nuclear war -- would really happen. The overwhelming consensus is it would not. Life, including human life, would manage to survive, barely. Even with the nuclear stockpiles held by several countries not all life could be killed off -- or so it's believed. This is probably so, but the point is to make sure we never have the opportunity to find out.

Hobnob is a dear friend and he lives in the vicinity of NYC, so I decided to spare his hometown.

The total end of human life -- actually of all life, although what happens in the oceans is left a bit unclear -- was never a realistic outcome. In the 50s (or Shute's imagined early 60s) there simply weren't enough nuclear weapons in the world to bring about such a result, irrespective of what radioactive elements they employed. It just was not possible. Even today it's impossible by any realistic assessment.

Shute wrote OTB seven years after he moved to Australia in no small part as a rebuttal to those -- and there were many -- who deluded themselves into believing that an atomic world war that engulfed the Northern Hemisphere -- which was and is a reasonably realistic scenario -- would have no effect on the South, which could go on in somewhat straightened circumstances but essentially unaffected. Akira Kurosawa based his 1955 film I Live in Fear/Record of a Living Being on this premise. Shute wanted to convey the fallacy of this thinking by conjuring a worst-case scenario that would scare people away from thinking about nuclear war as survivable, let alone winnable. But the best scientific assessments are that even such a catastrophe as a nuclear war would not go so far as to eliminate all life on Earth. Life will survive, as it has similar global disasters in the past. The real issue issue is how much life, and in what state? And a war in one part of the planet would not simply leave the rest of the planet unaffected -- it would have monumental environmental and other devastating effects even for areas spared direct combat.

That's the power of Shute's tale -- that it warns people that they hold the very existence of life on Earth in their hands, and to act wisely. The film makes this point too. But they do so by proposing a dramatic but at base unreal end game. Life would not be exterminated. But it would be driven to the brink, and take centuries to recover, with unfathomable difficulties and consequences. That wouldn't have been as easy or direct to set forth in the book or film, so Shute went straight for the jugular of total death and hopelessness -- sort of a nuclear "scared straight". In this it was better than posing the actually more credible result of some survivors, because even that scenario can still tempt fools into believing that nuclear war is manageable and "not so bad".

Shute fought delusion with illusion...with the caveat that no one really knows what such a war might bring, so that it's best not to try to tempt fate by finding out.