I was content to let it go (because: NPR) but now you’ve spurred me to look it up:
Narragansett turkey - Wikipedia
The Narragansett turkey is a breed of Meleagris gallopavo which descends from a cross between the eastern wild turkey (…
…which was no help at all, except maybe to point to English turkeys just being New World turkeys that at been on the other side of the pond for a long time. So yep, originally, all turkeys were from the Americas.