How does a guitar body affect electromagnetic pickups?
I think the best way to think about this is to not only think about the wire (moving guitar string) cutting through the magnetic field of the pick-ups (stationary) but also to think about the pick-ups themselves moving with the resonance of the wood that they are fixated to.
Now you must consider the frequency response of the instrument as a whole and this would definitely be affected by the choice of wood. How much it affects the sound is another question. I personally believe the subtle differences end up being audible due to the degree of amplification. But in the grand scheme of things, this is just a part of the entire signal chain that can effect your sound.
Of course it does not affect the "sound of the pickups", but the pickups don't have any sound to themselves anyway: they only pick up the sound that's coming from the guitar strings and modify it.
The sound of the strings, on the other hand, is affected quite a lot by the mechanical resonances of the guitar, in particular by the neck but also by the body. Because on a frequency the guitar can easily resonate to, the strings are able to release much more energy into the environment than for a frequency where the body does not move along at all. And which frequencies these are depends on a lot of parameters, which are different for every guitar type and even for guitars built in exactly the same way but from different types of wood.
So if you equip, for instance, a Les Paul and a tele with exactly the same pickups an circuitry, you will still get very notably different sounds. The difference is even more obvious when you also consider hollow-body guitars: quite a lot of these actually do have the same pickups and circuitry on them as Les Pauls, but still sound completely different.
To make the point about the mechanical resonances a little clear: if you built a guitar by spanning one string across a plastic broomstick, no matter what pickup you equip it with, it will always sound like just what it is. That's because the material is so light and soft that it can resonate to virtually any frequency and absorb a lot of energy.
On the other hand, if you use a massive block of granite, which is so heavy and hard that it will resonate but very little, you get a very long sustain (most damping is then due to string stiffness and perhaps even air friction) and a literally very hard sound, because none of the overtones are filtered particularly strong.
One of the major ways the vibrations can affect the sound of the guitar is by allowing greater feedback influence (most especially with hollow-body electric guitars). Example, Trey Anastasio has several extremely resonant guitars, and that controlled feedback is a signature of his sound.
As for actual EM influence in this case, there isn't any; but that doesn't mean the body of the guitar can't have a tremendous effect on the guitar's sound and playability.