In the event of thermonuclear war, very few people would die due to radiation exposure, and almost all of that few would die from chronic exposure causes, such as thyroid cancers that could have been treated had all the hospitals not been nuked.
Acute radiation syndrome would be rare, because thermonuclear weapons are very powerful, and fairly clean; Anyone close enough to get a lethal radiation dose from such a weapon is almost certainly going to die from thermal or blast effects long before they could develop any symptoms of ARS.
To see large numbers of ARS cases, you would need to fight your war with low yield fission bombs; Nobody's seriously planned to do that since the 1950s.
Disagree. In terms of direct exposure you're right, only small bombs can fry you. Big bombs kill farther away than they fry. However, the fallout is another matter. At the height of the Cold War the estimates were 2000 rads across much of the northern hemisphere. AFIAK we don't have good data on the lethality of radiation spread over time, but that much over a few days is most likely lethal. (Known data points: As a single exposure you have no chance of survival. At ~10 rads/day you only face an increased cancer risk.)
"The estimates"
According to estimates by crazy lobby groups, Chernobyl rendered half of Europe uninhabitable, and caused hundreds of thousands, even millions, of deaths.
Realistic calculations of likely radiation exposure from fallout after a large thermonuclear war suggest almost zero exposure to sufficient doses for ARS. If you were close enough to get a sizable dose, but survived the heat and blast, then you had to be sufficiently well shielded to prevent ARS in the first few days. After that, the radiation drops off very rapidly as the short lived isotopes are the ones that are active enough to be of concern in this regard.
You still have the problem of chronic exposure in the absence of modern medical facilities, but acute sickness would be rare.
Fallout plumes would mostly fall either on sparsely populated areas, or targets where there were few survivors other than those with shelters.
Low level fallout is just not that dangerous, despite the crazy claims of CND and Greenpeace.