Last year we were in the MK Christmas Day until around 1:00. The crowds were not bad the first couple of hours (7 am - 9 am) We got to do all of Fantasyland a couple of times and collected Fastpasses for the rides we wanted to do. We had BBB reservations and then CRT so we were out of the crowds from about 10:30 to 1:00. If you are staying on site and get there early MK is okay on Christmas morning. We did it two years ago too and due to rain it was even less busy than last year.
We had no problem getting into Epcot for our dinner reservations at 5:00.
Here is the info that I saved from last year that listed which parks and when they closed during Christmas week.
Last week was extremely busy at all of the Walt Disney World parks, as the Magic Kingdom began phased closings on Monday (25th), Wednesday (27th), Thursday (28th) and Sunday (31st). Disney's Animal Kingdom also closed on Wednesday (27th) and Thursday (28th), while the Disney-MGM Studios had parking lot overflows on Wednesday (27th) and Thursday (28th), as well. Epcot, with its larger parking lots and much larger capacity, did not close at all. It's uncertain when three parks last closed at the same time.
While Epcot did stay open, lines were extremely long, with waits for Soarin' regularly around 3 hours and lines for many Future World attractions running all the way to the Innoventions breezeways. New Year's Eve saw Soarin' waits of up to 4 hours.
Hope that helps.