you know i highly doubt that he would have an accent. there are lots of actors and actress's that have accents , but they dont use the accent in the movie they play in. Such as Kate Winslet in Titanic, she has a british accent, but yet, she didnt have an accent in the movie.
I think as long as the movie doesnt call for an accent there is no need for one.
all in all i dont think Robert will have an accent because it doesnt call for one. And Edward doesnt have an accent just like you mentioned
=)

He's using an an American accent and apparently it's "faultless". I've that the only difference is that he speaks a little more properly and clearly due his normal accent and that fits the character of Edward perfectly. He was supposed to talk a little differently anyway..pronounce his words a little more carefully, since he's from 1900.

hes actually got a pretty good american accent. if you watch some of the videos of him talking, he sounds great.

When they first announced the movie, one of the first interviews with Catherine included the fact that he has a perfect American accent.