Depends on if it is an air or ground burst and the specific distance from the burst.
In
most scenarios, a high altitude burst is on the "unlikely" side of things, so lets assume you are talking about a low altitude air-burst which produce Source Region Electro-magnetic Pulse [SREMP]. This is mostly a radiative effect, as well as charge buildup due to seperation of air molecules into charged ions. Low altitude of course is relative, since depending on the magnitude or the weapon in question, and the terrain of the target area, the most efficient burst height is variable. Hiroshima and Nagasaki were in valleys which contain and magnify both thermal pulse and blast effects, so their fuses were set for fairly low altitude (~200M). Modern weapons have much higher yields, so the burst height is much higher.
Let us assume then that you are speaking of a 1 megaton blast at a 2 KM altitude at a distance great enough such that overpressure is not a significant concern, or at least will not throw the aircraft outside of recoverable flight conditions.
The radiative effects of a nuclear weapon fall off quickly past the immediate blast radius. Gamma rays are quickly absorbed by the air. Significant Ionization effects in the atmosphere are limited to approximately 10 KM. (Radiation sickness and cancer is caused primarily by irradiated dust---Fallout)
EMP can be generally split into 3 sections. E1 is Compton radiation created when gamma rays knock electrons out of the magnetosphere. (Not a concern in low-altitude bursts) . E2 is the result of ionization of nitrogen molecules, and is very similar to the EMP produced by lightning. E3 is the result of the earths electromagnetic field being displaced and then coming back. It is similar to EMP caused by solar wind.
The skin of aircraft to some extent acts as a Faraday cage (Assuming it is not composite). However, this is of no help to radio equipment, and it is not a very good Faraday cage (windows, I mean, your cell phone works does it not?). Damaged radio equipment is of little concern, since radiation trapped in the ionosphere would make hash of any radio signal anyway.
What is of concern is the control surfaces electronics, and those regarding engine regulation. They generally don't have a whole lot of long exposed wires, and are protected by copper mesh. These systems are designed to be protected against heavy solar wind produced by solar flares as well as lightning strikes. Since I have essintially just described the E2 and E3 pulses produced by a low altitude nuclear detonation, my final verdict is that if the EMP is large enough to effect critical subsystems, the aircraft has other things to worry about.
According to the third edition of The Effects of Nuclear Weapons, by S. Glasstone and P. Dolan, the threat to electrical and electronic systems from a surface-burst EMP may extend as far as the distance at which the peak overpressure from a 1-megaton burst is 2 pounds per square inch.
An overpressure wave with that much power is probably enough to send most civilian aircraft into an unrecoverable spin.
Nuclear Weapon EMP Effects