Just a point of clarification. Those who suggest wood influences the tone of an electric guitar are not suggesting that wood has any direct influences on the magnetic properties of the pickup. They would suggest any tonal differences would be due to the wood absorbing and/or reflecting vibrational energy of the strings. The idea being, different materials tend to absorb and reflect energy differently from each other. So, any tone differences would be due to the resultant vibration on the string, itself, that the pickups detect and convert to electrical energy. If this were the case, we would expect to see any difference within milliseconds after the initial attack. Though, at the instant of attack, there would likely be no difference. It would take some time, albeit very brief, for any differences to become apparent as vibrational energy is either absorbed or reflected.
That is the theory, anyway.
Personally, I suspect that if such a system could be subject to truly objective testing using laboratory test equipment (not hearing tests), there may be some measurable difference. Whether that difference is able to reach the level of detectability of human hearing is a different matter.
To illustrate, one of my sons has his master's degree in mechanical engineering. He recounted an interesting story of an open-house his place of work had recently. The idea was to win a prize if you could guess the weight of your own signature. That's right. He works around test equipment so sensitive it can detect the difference in weight of a piece of paper before you have signed your name and after you've signed your name. Of course, no human being can detect something quite so minute. That's why I say I do suspect different body and neck materials may, in fact, affect the vibrational characteristics of a string that may actually be testable and measurable on professional, laboratory-quality test gear. But, it's also possible any such differences may not rise to the level of really mattering to what we hear.
....Or, they may...or may not!