The cold can defo make the throat muscles stiffer. The initial cause of stuttering can still be in the brain (i.e. anxiety) of course, but this neurological cause manifests itself in the throat muscles. There is a video that shows what happens with vocal chords during stuttering: they are viciously pulled apart, and get into cramps. But we are doing this with the throat muscles indirectly. So probably the throat muscles are getting into cramps, which then pull the chords into cramps too. So: if the throat muscles are already tense (for example because of a lot of speech therapy, aka extra vocalizing, throat muscle usage), then this fatal cramping happens way easier, stronger and longer.