Is gamma and beta radiation a serious threat in nuclear war?

Megatonnage detonated is (almost) compltetely irrelevant - the design of the bombs and where they are detonated will define the amount of radioactive material being deposited into the biosphere. For example, a large bomb, detonated for its EMP in the stratosphere will have very little impact, a much smaller bomb in a shallow surface detonation will be devastating.

Next - you know where "radiation" is coming from in this case? Instable atomic nuclei splitting. The half life of these nuclei will depend how long the radiation stay around, that can vary from seconds to several 10,000 years. The type of the nuclei (or rather, the atoms, new we're going into chemistry) will depend how these will circulate through the biosphere.

For stuff outside of your body, basically only gamma radiation is intersting - alpha and beta dosn't have the range to reach anything interesting as long as there's a few meters of air between you and the radioactive material. However, that's not the big problem - the big problem is the radioactive material coming into contact with your body or entering it ("incorporation").

In the former case, a thorough shower usually gets rid of it. The latter case, though, is a problem - especially, if that material is chemically either identical or very similar to elements used in biology, like Calcium or Iodine. That stuff is then absorbed by your body and used - like Iodine in the Thyroid. In these cases, Alpha is the most dangerous type of radiation, because it is directly absorbed in the first few tenths of millimeters in your body, wreaking massive local damage.

Third possibility, as some already pointed out, is fine dust. You can shower it off the surface, sure, but once you breathe it in or ingest it, the fun (not!) starts in Earnest. Similar to the above - doesn't really matter, in the end, whether your lungs get irradiated from Plutonium dust you can't get rid of in time, or your bone marrow gets destroyed by the radiation from ingested Caclcium which has been deposited into your bones.

No. Alpha radiation can be stooped by a sheet of paper. Alpha radiation can be very damaging in close proximity (for example, if you inhaled some plutonium dust) because of its huge mass, but it has no penetrating power.

Beta radiation is free electrons. More dangerous, but nothing compared to gamma. The gamma radiation is what is going to have the highest lethality.

Some bombs are designed to produce a large burst of neutrons. Neutrons are also massive, and can do a lot more damage than alpha particles.

Edit:
I failed to mention that the neutrons are what cause radiation to "hang around". High energy neutrons can "activate" other elements. Essentially, the neutrons bang into another atom and manage to stick in the nucleus, changing normally stable atoms into unstable radioactive ones.How long these neutron activated elements stick around is a function of the element that has been activated, not the neutrons themselves

There are many dangers in such a war. the answer sought is too simplistic.
Once the initial explosion is passed the dangers are in the products.
There are countless radiactive and poisonous substances created.
Now if these were solids in the soil then the risks can be alleviated by staying away from the place.
The airborne dusts are the biggest risk.

Now if you breathe in some of these dusts or even worse they enter the bloodstream then the alpha particles are the greatest danger. Because they are inside you and they interact strongly with anything that is near.
Beta particles can travel a greater distance but it also means that many beta particles can pass through you without interacting.

The various isotopes have different half lives. It isn't that alpha, beta or gamma radiation hangs around. It doesn't.
It is the various atoms that hang around.

Short half lives give a high dose of radiation but they disappear.
Long half lives are basically around for ever but the amount of radiation per year is low.
It is the intermediate ones with half lives of months to decades where the greatest risk lies.
Active enough to be deadly, last long enough that you cannot shelter for long enough to be safe.

But basically why worry. In a nuclear war the lucky ones die quickly. The unlucky ones live longer and die from a) crime, b) lack of food, water and resources, c) radiation d) poisoning.

A long time ago we realized that an all out nuclear war is unthinkable. It is mutually assured destruction.

Alpha is the weakest of the three, it can be blocked by a piece of Aluminum foil. "hangs around" ? radiation doesn't hang around, it is emitted by the source and moves off in a straight line until absorbed by something.