I have gone to the World 2 years in a row the week after Thanksgiving. It was thought to be a down time in years gone by which is part of why I chose it. I can tell you that it is nowhere near as busy as Turkey week-end however it is a far cry from low crowds. I still love it though. Warm enough to swim, Christmas decor up, Christmas parade taping in MK. Just lovely - which is probably why it's getting busier.
ETA: If you are talking about the Saturday immediately after Thanksgiving (which is Thursday I think?), then it will be crazy busy. But, if you are talking the Saturday the week following, that will be parade taping. Very, VERY busy at MK.