I think either Epcot or MGM is the way to go with an extended family such as yours. Yes, each country celebrates NYE at the same time as it is celebrated back home. For instance, Italy celebrates at 6pm .
I mention MGM only because they will be showing Sorcery in the Sky fireworks that night. They only show those particular fireworks 2 or 3 times a year and NYE is one of them .
I think all the parks are equally good on NYE. It just depends on if you can get in or not. When we were there in 2000 for NYE you had to go to the park first thing in the morning to get a hand stamp in order to come back that night. I'm not sure if that is still the rule. We ended up doing Pleasure Island for NYE 99 and MGM for NYE 2000. It was fun but too crowded and insane for my kids.