Given that the movie takes place during the reign of Richard I, all the characters should be speaking Early Middle English, something of a transitional period between Beowulf and Chaucer. With this in mind the others actors' modern English accents aren't really any more "correct" than Costner and Slater's American ones.
Many linguists theorize that the American accent actually is the original British accent.
Furthermore, as part of the Anglo-Norman aristocracy, Robin may not have spoken English: Norman French was the language of the English nobility of the time, and in fact Richard himself didn't speak English at all.
The Sheriff and his cousin, Guy of Gisbourne, both eye-rolling when the Bishop asks for God's blessings on the Sheriff? The Sheriff is a devil worshipper (and so is his cousin, possibly.)
In the extended version of the movie, we learn a little about the Sheriff of Nottingham's parentage. The witch Morgana herself was his mother, and she killed a baby in the palace so that she could substitute her baby in his place. This brings up a number of horrific implications.
Probably one of the most subtle is shown when Morgana talks about how she has been seen as a freak all of her life. Given that, how did she become pregnant? Given her lack of sympathy for Marian's plight, it is likely that the Sheriff is the product of rape, perhaps even rape by a nobleman... in which case, having her son produce the heir to the throne through raping a nobleman's daughter is the ultimate act of revenge.
This also explains why the Sheriff, upon demanding a "proper" wedding to Maid Marian, snarled that "for once in his life he would have something pure".