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.
The most likely domain of interface for a mathematician with engineering would be in the realm of "computational engineering," in which numerical algorithms are used to study physical and engineering systems. This work is highly interdisciplinary, and requires close collaboration among all of the different researchers involved, as very few are experts in all of the areas with which they are concerned.
However, there is a certain amount of logic to the idea of staying in one's "home" discipline. It makes it easier to figure out "who you are," which can make starting your career a bit easier. One possibility might be to pursue a degree in applied mathematics, in which you study problems relevant to engineering, rather than remaining in "pure" mathematics. This might be a reasonable compromise that keeps you in the mathematics field, while still allowing you to pursue topics in areas that interest you more.
I have been in your shoes, as I migrated from mathematics to computational engineering. While my math background gave me some advantage, I remember it was difficult to make the transition because I was very unfamiliar with a lot of the key concepts in engineering. What really helped me was watching online courses (MIT, IIT, etc...) in the material I was missing... Of course, I had an entire year to prepare for it. If you have a lot of time to make the transition, by all means do so and you shouldn't have "too much" trouble. But if you're pressed for time to make up for a lack of engineering background, it can be quite overwhelming. It's best to hit the ground running in graduate school.