I have been researching this for the past hour or so, but I have not really found a sufficient answer. What I have found is that vaguely, math education PhDs set policies for math education. However, universities almost uniformly have math professors teach mathematics courses. I believe that usually the math professor runs his course the way he wants to. Do their policies affect the university level, and if so, why do math professors teach the courses instead of math education professors?
Someone (probably the salesman) says: Student clickers! the greatest thing for low-level large lecture math classes since the invention of the overhead projector! So, the math education PhD (Professor Z) sets up a study. Some classes done with clickers, others without. Pre- and post-testing, to see what the students learned, and the students' impressions of the two methods. Publish in a math education journal. Get kudos from Professor Z's department (maybe a raise or a promotion). Whether it will influence anyone in deciding whether to adopt the use of student clickers: that's another problem.
Having a PhD in anything doesn't suddenly make you into anything. Lots of people with Mathematics PhDs don't work in universities and don't do mathematical research.
People who have a PhD in mathematical education will have spent time researching mathematical pedagogy, more often than not, at pre-university level. In the UK, people who have done this are then often involved in teaching and training those who are training to be a school teacher.
Academic reform, e.g. introducing new curricular, is often done by panels of maths teachers, maths professors, politicians and educationalists.