Great Northerns will work just fine. So, too, will Navy beans, although they're smaller than either of the other two.
FWIW, all common beans are interchangeable in any recipe. You will get subtle flavor nuances, one to the other, but not enough to matter. About the only time it makes a difference is in a recipe where the color of the bean is an integral part of the dish.
While all beans have subtle flavor and textural differences, most of the time they are pretty interchangeable, unless color is an issue. Will the dish come out exactly "right?" No, but normally it will be close enough not to get too hung up on having the exact right bean.
Just to pick a nit, Pete, you really should specify common beans. Old World beans, such as favas, have a distinct flavor profile that's different and identifiable.
This would apply to a very few common beans as well. Black Turtle, for instance, can't be mistaken for anything else. But, in general, it's true: almost all of the 10,000 known common beans can sub for each other.
I would not, however, substitute limas or cowpeas though, as they, too, have distinct flavor profiles that could change the recipe dramatically.
It's interesting, too, that the OP used "white kidney" as a synonym. In the 18th century, and well into the 19th, "kidney" was used generically to describe all common beans.