I haven't seen "Dumbo" in ages and I've never heard of "Uncle Remus", but I think when trying to determine if these movies are prejudiced or not, we have to keep in mind when these movies were produced, and the social atmosphere when they were released. Much of the material produced before the time of political correctness going overboard can be seen as kind of racist. I don't even like Western movies b/c of how they portray Native Americans, and don't even get me started on "black face".
Just like when you were a child, you didn't notice these things, I don't think our kids will. I say let them be children and enjoy these characters without the judgement of adults. Of course, if they have questions, answer them honestly and in an age-related way, but I don't want to bombard my child with worries thinking every little thing is racist.