Tyson’s stutter comes from his anxiety rather than a speech impediment and it almost entirely social.
He is comfortable around the Moriarty Mirrors and The Colonel, even though he shouldn’t be for a variety of reasons, and due to this Tyson doesn’t stutter around them. Tyson doesn’t stutter when addressing his brother or his dogs; or animals in general, actually.
The more comfortable Tyson becomes around a person, the less he stutters around them, and the more confident his speech; as he drifts away from clunky, stilted sentences to the words he actually wants to say yet avoids to lessen the severity of his stutter.
Tyson has dozens of prepared phrases in his back-pocket and part of that is about avoiding his stutter. When saying a line he’s rehearsed for hours, Tyson is incredibly unlikely to stutter. That is why he typically introduces himself with the familiar, “Lieutenant Tyson Delaney, at your service.”
In many ways, those phrases are a crutch. If he uses them too much, Tyson doesn’t truly get comfortable around the person and - as such - his stutter doesn’t reduce.
His stutter changes based on who he is addressing. For example: when talking to Jim with a stranger in the room, Tyson will not stutter when addressing Moriarty and the room at large, but he will stutter when directly addressing the stranger.
Being surrounded by crowds on the street will also cause him to stutter around someone he otherwise wouldn’t, as his anxiety overrides his comfort zone and calm.