Well, that’s just completely wrong.
First of all, there really isn’t all that much radiation.
Second, radiation doesn’t interfere strongly with radio, because they are on completely different wavelengths.
While it’s true that a nuclear bomb can create radio interference, this is not because of the radiation as such, but because of a single huge blast of radiation covering all frequencies at high energy, and it’s a relatively short-lived phenomenon.
For reference, FM radio is in the range of 87.5 to 108.0 MHz; AM radio is 150 KHz to 26 MHz, roughly, and gamma radiation (“nuclear radiation”) is in the neighbourhood of 10 EHz, or 10[sup]19[/sup] Hz.
For the same reason that radio channels don’t interfere with one another, gamma radiation does not interfere with radio signals.
Of course, all of that is rendered moot by the first point: There’s not really a lot of radiation around in Cataclysm.