Graduate math student migrating to engineering?
The only problem with switching to a particular field of engineering is that you'll be missing a core set of knowledge particular to that discipline that will hinder your progress for a while.  For instance, you would probably do great in many electrical engineering sub-diciplines, but if you've never taken a circuits or signals course, you're going to be lost for a while before you can learn that material.

If you already have research experience, that will count for a lot, but I would suggest finding an engineering program that would allow you to take a few undergraduate courses during your first year or two in order to get up to speed.  If you find a professor that is looking for a mathematics-savvy graduate student to fill in the gaps in his/her lab, that might be the best way to get your foot in the door.  You've got skills that could be very valuable to the right lab, and if you use those as leverage to join a particular lab/research team, you'll be able to fill in the missing pieces to your new field and move forward with your own degree.