dore_m If you play a note on an acoustic, hold the note, and then push the pedal down - does the timbre change?
I assume by "pedal" here you mean sustain, but just in case you meant sostenuto or una corda, my answer is "definitely not", because nothing that could have changed the timbre is happening. It equally won't change if the piano has a "soft" pedal (typically on an upright), either by moving the hammers closer to the strings or by putting a piece of felt between the strings and the hammer (actually the latter might marginally change the timbre by selectively screen some frequencies, but we're spitting hair with that).
Now, with that out of the way, if you do mean sustain as I believe you do, the question is what do you mean by timbre? I can definitely hear string resonances if I push the pedal down without any note playing (the "famous slushhhhh"). If I do the same while a note is playing, I can definitely hear the resonances, and hear them stop if I raise the pedal, and start again if I push it again. I've just tried it right now on my grand. I am not sure I'd call that "timbre" though. It's more like "ambience".
And finally, to state the obvious, if the pedal is lowered before the note is struck, the effect is much, much larger. That I can definitely call "timbre". I believe this is due to the much higher harmonic content during the attack which can excite a lot more of the other strings than the sustain can if I lower the pedal at a later time.