Reaching back in my memory....
We took DD with us in early '04 and from what I remember -
MGM - had a completely private curtained area (similar to a large dressing room) where I could nurse DD. I believe there was a glider rocker in each, and I think there was a seat where DH could sit. I remember there being only 2 of these private areas/spaces.
MK - had a little room where there were about 6 or so gliders/rockers (that's an estimate), so I guess you'd say it was a semi-private room? It was dark, and quiet, and there was enough space between the rockers that I didn't feel uncomfortable with the other moms & babies in there when we were. Not completely private, but comfortable.
**ETA - I did plenty of nursing around the parks too as it wasn't always convenient to hike back to the baby care centers when it was feeding time. I usually stopped there on the way in and out of the parks, but otherwise just found a quiet, secluded area (plenty if you look around) where I could cover up with a blankie and nurse DD. I was very discrete, and always felt like I could find an out of the way place where we were both comfy (and I wasn't offending anyone).
Didn't do Epcot or AK that trip so I can't comment on those parks. Hope this is somewhat helpful!