revealed himself as the Klingon Voq (STD:S01E11:The Wolf Inside) he began speaking with the voice of Voq (which we heard in earlier episodes).
This is never directly answered, but Shazad Latif, the actor who plays both Tyler and Voq, gives us a clue in a 2018 interview. He mentions incorporating some of Voq's speech patterns into Tyler's voice, and comments (emphasis mine):
Latif: I always wanted to have a hint of Voq in Tyler. So his American accent always has a kind of pharyngealness and I wanted to keep that up with the English when he's doing the Klingon. There’s so many different ways the Klingon throat would work. It’s got a bit of that Arabic sound.
Latif suggests that a large part of the Klingon voice is actually in the throat, rather than in the vocal cords themselves. This isn't something human physiology necessarily struggles with1: as Latif notes, these sorts of sounds are common in human languages like Arabic, and are sometimes employed in singing.
If this is the case, then it's most likely that Voq is simply using the muscles he's used to speaking with, which produces a more guttural, Klingon-y sound than the muscles Tyler is used to speaking with. The vocal cords barely need to enter into it.
1 This should be obvious, since humans are self-evidently capable of voicing Klingons in real life