Which of the response is faster? voice or hand action?
Assuming that we are talking about a situation at the physical limit of the brain:

Speaking a word will always be slower than pushing a button.
A whole word requires a contraction of the diaphragm to push out air so this air can be modulated by another sequence of contractions and relaxations of the vocal chords. 

So contracting finger muscles is very similar to speech initiation however for a computer to respond (in the example you were playing a video game) the computer needs to recognize the full word and then map it to an output before actually implementing the left turn.

So adding the extra processing time of the computer and the actual extra production time of the vocalization you're better off pushing buttons.

As a matter of curiosity if the physical limit situation is relaxed; whenever there is a mismatch in both sensory receptors as well as outputs, the brain does its best to compensate. Think of something dropping on your foot, the visual information is actually delayed so that the sensory stream (that has to go up all of your leg and spine) catches up.