Are nuclear bombers themselves damaged after dropping bombs?
Mostly, no.

The number one factor protecting the aircraft from the effects of the explosion is distance. The aircraft is traveling at a sufficient speed to be very, very far away from the bomb before it detonates. All effects of the explosion can be multiplied by the fraction: 1 / r, where r is the distance between the bomb and the bomber at donation (because circumference is directly proportional to radius). While the effects may be great at the target, they spread out as the circle grows.

Detailed Analysis:

As you emphasize the speed of light of radiation from nuclear reactions, I would like to write something about radiation and how it could influence the aircraft.

Radioactivity

Nuclear radiation consists of gamma rays, which are some sort of EM waves and particles like helium nuclei (alpha), electrons (beta) and neutrons (without Greek letter...). There is also some more stuff like positrons, protons or heavier fragments of the nuclei, but they behave similarly.

The particles indeed escape with a large fraction of the speed of light, but this has nothing to do with the nuke. They will escape with the same speed from a nuclear reactor or from tiny probes in a lab. The nuke just spills more particles than the probe, but does not give them more speed.

Alpha &amp; beta radiation

You may know you can stop alpha particles by a piece of paper and beta particles by a 3mm sheet of aluminium. In air (ground, room temperature), alpha particles can travel not more than 10cm, and electrons not more than 10m. The exact range depends on the energy of the particles, but these are the maximum ranges for nuclear reactions. Also, the distances depend on the density of the air, so they travel farther in hot air / at high altitude.
In general, there is a high electric field inside atoms, and the charged particles.

Finally, these two particle types will never reach the aircraft throwing the bomb. Even if they could, they would be stuck in the hull and not do any damage.

Neutrons

While alphas and electrons are absorbed on a short range because their charge interacts with the strong fields inside the atoms they are traveling through, neutrons are neutral (really!), which makes them very special. They like to travel through heavy materials like lead, but are easily stopped by plastic or water (one reason for that large swimming pools in nuclear power plants, though swimming usually is prohibited). The reason is that neutrons are like ping pong balls, and lead atoms like basketballs. The basketball will not notice when it's hit by the ping pong ball, while the ping pong ball flies ahead with the same velocity, but different direction. The next hit(s) with other basketballs may put the ping pong ball into its original direction, still with the same speed.
Hydrogen on the other side is a atom as light as the neutron, hence it's like resting ping pong balls. When hit by the neutron, a hydrogen atom is punched back and absorbs a large fraction of the neutrons energy, causing it to slow down. If the hydrogen atom is bound in a molecule, it may be separated from it during this process. That's why plastic is a good shielding material, but nitrogen and oxygen in the air are already too heavy atoms, so air is a bad shielding material.

The metal the aircraft is made of is (nearly) not affected by neutrons. Unfortunately, almost every molecule of the human body contains hydrogen, especially the DNA. While the pilots do not suffer imminent injuries from radiation, this may increase the chance to get cancer. 

However, Paul Tibbets, pilot of the Enola Gay (Hiroshima bomb) died in the age of 92.

One fact: "Neutron bombs" are nukes optimized to release a gigantic amount of neutrons. When ignited at high altitude, it kills all animals and humans, leaving the infrastructure (and almost all plants) intact. It's good that they are proscribed by all nations.

Gamma radiation

Coming to gamma radiation. People are most afraid of it, as it penetrates anything, and you know, the Hulk was created by it. Its nature is that all the photons it's made of have a tiny chance to interact with the atoms they are passing by. A few photons are stopped, the rest go ahead. This is why each centimeter of material reduces the intensity by a certain factor, but you will never shield gamma range completely (mathematically). Typical gamma rays need some 10cm of aluminium to reduce the intensity to 50%, or less than 10cm of lead. So, it will pass the aircraft's hull and also the pilots. Again, it will not influence the aircraft. And because it only reacts here and there with matter, there is a high chance the pilot's body will repair the damages, and the pilot will not turn green.

Other effects

Shock wave

As said in the other answers, the explosion causes a big shock wave which will definitely shatter and possibly damage the aircraft. Wikipedia says about the Tsar bomb that it was released from about 10km altitude and detonated at 4km. Using a parachute, the bomb was slowed down allowing the aircraft to travel 45km after release. Yet, the aircraft fell about 1km when it was overtaken by the shock wave.

Edit: The shock wave travels at the speed of sound, so the faster the aircraft, the farther away it is when it's overtaken by the shock wave, and the lower the intensity is. (And there are supersonic aircrafts... However, the Tsar shock wave traveled around the earth three times)

Heat

A nuclear bomb releases a lot of heat. It is said that in Hiroshima, trees 10km away from the explosion caught fire. But the aircraft was farther away and its hull is not only well cooled, but also much less inflammable than a dry tree in the summer. Nevertheless, the "Tsar-bomber" was painted with a special heat-reflecting paint according to the Wikipedia article.

EMP

Each nuclear bomb also releases a electromagnetic pulse, which can damage wide electric circuits like power grids as well as any semiconductors. However, the hull of an aircraft already provides some protection against this, and the protection can be improved more to make the aircraft EMP-hard. While today's bombers should be immune to EMP due to these protections, I guess the ones which actually have thrown a nuke were immune as they didn't use any semiconductors.

Summary

It turns out that radioactivity is not an issue because not all parts have the range to reach the aircraft, and those which do, have no influence on the aircraft. 

The highest thread comes from the shock wave, which definitely damage the aircraft or at least bring it into an odd situation. I don't know what an issue heat is, but the Tsar bomber had some heat reflective painting, and an aircraft usually has a good cooling by its environment. EMP is not an issue as military aircrafts are usually already protected against them.

Remarks

Of course, radiation damage / effects depend on the time something was exposed to radiation as well as the intensity. So, when I wrote gamma rays will not harm the pilot that much, this already includes a larger distance and thus lower intensity. Near the detonation, the gamma radiation alone would kill you.

Being exposed to radiation does not automatically make an object radioactive itself. Gamma rays do not make anything radioactive, they ionize atoms and may destroy molecules. The particles may be captured by atoms forming new isotopes / elements. This isotopes / elements may be stable, or may not. If not, that atom is radioactive. However, for most materials, this induced radioactivity has a short decay time.

As said, alpha particles are not an issue as shielding against them is easy, they even can not penetrate the skin. Harmless? Body cells are most vulnerable to radiation damage during their separation, and your guts are covered with mucous membrane, where this happens very frequently. Swallow a piece of alpha emitting material, and getting cancer is more likely than just possible.

You see, radiation is a vast topic, and what I wrote about aircrafts and pilots may be correct for them, but not in general. Their main protection is: Distance is the best shielding, the farther away, the better.



Hmmm, this answer got a bit longish... sorry for that.