It is certainly possible to get a masters in engineering after undergrad in applied math, and your Masters is what would matter most when looking for a job. But you want to be as prepared as you can be. I suggest you take a look at prospective masters programs that you may be interested in, and make sure that you do what you can to take relevant courses so you are prepared for masters level work.
For Mechanical Engineering I cannot say that I would know (I am electrical), but perhaps intermediate mechanics and thermal physics courses from the physics departments would help. These may be adequate substitutes for the required soph/jr level mechanics and thermodynamics/heat transfer courses that undergrad mechanical engineers usually need to take. there may be other obvious classes you could take as well.
Some areas of electrical engineering are essentially applied math - signal processing, information theory, and controls comes to mind (you can do controls in Mechanical engineering, too!). Applied math would be a great background for these. For EE it would help to take an intermediate electrodynamics class and electronics from the physics department, and learning as much about Fourier analysis as you can. Your computer science department may off courses in logic / computer hardware design. These would be useful for EE, too.
Finally, all engineers need to learn probability / statitics and computer programming. Numerical methods can be good, too. I am guessing you get plenty of that in applied math?
In any case, your faculty advisor is likely a great (perhaps the greatest) resource to help you make the most of your current opportunities. Speak with them about your goals and take their advice seriously.
I wish you the best,
jason