This is a universal trait. French speakers have told me they can tell an American speaking French from a Brit speaking French, by the accent to the French being spoken. I have German friends whose children grew up here in America, and when they moved back to Germany, the children were ridiculed by their friends for speaking German with an American accent. You are correct in that this is a Book of which we speak. You can't really write dialogue so that the difference between a Mexican speaking English and an Indian speaking English is apparent.
Writers working outside the boundaries of a completely realized fantasy world do it quite easily.
It is perfectly reasonable to assume that the different races spoke Westron in slightly different ways, depending in some part on what language they spoke normally. And given that the languages are based on different European languages Tolkien studied, it seems reasonable to assume that the Westron spoken could have trace elements of other European tongues. Bottom-line is that the movie will be filmed with persons speaking variants of English. Some might even be foreign speakers of English, who will bring their own accents to their roles. If this is going to ruin the film for you, stay away.