Why doesn't the bomb cause radiation poisoning to Gotham City?

This is an interview with Edwin Lyman, PhD, scientific director for the Nuclear Control Institute, in November 2001.

Member Question: What are the levels of radiation risk at 10 miles, 100 miles, 100,000 miles from ground zero? How far can the radiation travel? Is that based solely on wind direction?
Edwin Lyman: Well, again one should distinguish between a nuclear weapon attack and an attack on a nuclear plant.
For a nuclear power plant accident, the acute effects I described -- acute radiation sickness with a high chance of death -- would be limited to within 10 miles or less of the accident. This is partly the basis for the choice of 10 miles as the size of the emergency planning for U.S. plants. However, the plume would be carried by the wind and depending on atmospheric conditions could contaminate a large area and even hundreds of miles downwind there would be some significant contamination. This was seen after the Chernobyl accident, where large sections of Northern Europe did receive significant contamination and the aftermath was detectable all around the northern hemisphere.
The range of effects from a nuclear weapon is very strongly dependent on its yield and whether it was detonated on the ground or at some height above the ground. If we take the example of a terrorist crude nuclear device, let's say one-tenth or one-fifteenth of the Hiroshima yield (which would be about a 1,000 tons of TNT equivalent), the blast affects would be fairly limited in range, probably to less than one mile.
The direct radiation effects would be felt considerably farther and fallout would travel hundreds of miles, and that would be directly related to atmospheric conditions and wind direction. So, the extent does depend on a lot of factors, but the ultimate effect could be felt tens or hundreds of miles away. However, the farther one is from the site of detonation, the more time there is to take countermeasures.

The bomb in The Dark Knight Rises was not a 'bomb' exactly. It was a nuclear core built for a reactor, converted to be used as a bomb by Bane and his followers.
Bruce Wayne has 2 minutes to fly 'The Bat' away from Gotham. Though there were words said between this and when he actually got in the cockpit, let's just assume he had the 2 minutes.

BATMAN: Two minutes. I can fly it out over the bay...

This article says that 'The Bat' flight sequences were done by helicopters, so we can estimate the speed of The Bat with that.

According to Wikipedia, the fastest helicopter(checked 11/28/2012) reached 250 mph in 1986, or a little over 4 miles per minute. With modern engineering and the resources of Wayne Enterprises and with the goal of building one aircraft, not a large fleet of cost-effective aircraft, I think the speed would be able to get a little further.

I am not convinced that Batman was able to carry the nuclear core far enough in the time provided to prevent the damage. We don't really have enough information about the capabilities of "The Bat" or the specifics of the nuclear core depicted in the film. There are also other factors that decide how far the radiation will travel (and over what period of time), and I don't know how heavily nuclear fallout has been researched. I think this is just something we have to accept as a 'given' for the purposes of the film.

Also, the film ended right afterwards, so there may very well be some radiation poisoning that was just not shown.