Since I'm almost always attending the parks solo, for many years my tradition was to visit Epcot early, then have Thanksgiving lunch at the Garden Grille. They serve family style, and while the menu would vary from year to year, they would usually put a holiday spin on this meal. They grow the veggies right there on property, and you can taste the freshness, it's so fabulous. It was a character meal, so there was always some interesting interaction with servers and characters, plus it's always a delight to watch families with children getting to meet with all the characters. I was always treated well there.
It's been my understanding the last few years that the lunch is no longer available at Garden Grille, so unless something has changed, that is probably no longer an option. However, another poster mentioned Liberty Tree Tavern in MK, another family style dining option. I've eaten there at non-holiday timess, and I'd imagine they would do a terrific job of presenting a traditional Thanksgiving experience.
I've always enjoyed Thanksgiving at the parks. Even solo, the atmosphere there is warm and inviting, and even if alone, you're not really alone there. Being grateful to be there helps me to remember everything else I'm thankful for. And there sure are lots of hugs to be had at Disney!
p.s. to OP. I've been blessed to meet a great many visitors from the UK over my years at the parks, and it's always been a delightful experience. Welcome, and have a wonderful holiday here!
but since I am not a big fan of turkey etc I try to avoid that at the parks but to my understanding Liberty Tree Tavern does a great T'Day....
Planes, trains & Automobiles comes to mind 
