The impact probably depends on many variables just like everything else in life.
What was the turkey/bunny population prior to their arrival? What's the habitat look like? After a few years they will reach the new balance point and then it's up to us in how we react and change things to how we want them.
It's like coyotes at my place. They are there and i hear them at least once a year. I have more success hunting then by accident when hunting deer or turkeys then when i actually attempt to hunt them. The foxes were nonexistent for a few years but now are making a comeback.