I'm a Physicist (studies Acoustics of musical instruments in college) and a bass player, so take my response as being affected by that.
The output of a magnetic pickup comes from the relative motion of the string versus the pickup - if the string moves, the pickup produces an output. If the body moves (the pickup is screwed to the body), the pickup produces an output. If both move, then it matters if they're moving together (lower output) or in opposite directions (more output).
It's obvious that everything on an electric bass moves. When you put a Snark tuner (it's an accelerometer, it senses physical vibrations of a solid body) on your headstock, the reason it works at all is that the headstock is vibrating. So the wood that the bass is made out of does vibrate, and it matters to some degree what it's made out of. The only question is to what degree.
On to the original question...
An underwound pickup has more bandwidth than a hot one one with a lot of turns. This gives it more bandwidth when loaded by the cable capacitance. Does that mean that the wood has more effect? Well, having studied the vibrations of different parts of musical instruments, what you find is that the vibrations of the bigger parts of things (like the body of your bass) are concentrated in a limited bandwidth - mostly low mid and below in the case of a bass guitar. I think you'll find that even the high output/more colored pickup has sufficient bandwidth to capture all (or almost all) of what's going on in the body vibration department.
Yes, the underwound pickup will sound different, but I think the differences in sound between that and an overwound one are more electrical than revealing more acoustical stuff - whatever role the body wood plays is there with both pickups.