I'll bet the tuners won't fit the other neck. They are replicas of the original Fender "backward" tuners and are designed to SAE specs while the Schallers are metric. "Backward," by the way, refers to the fact that the knobs are rotated in the opposite direction from most guitar and bass tuners. In other words, to sharp a string, the key is turned clockwise when looking at it from the side.
I assume the neck came without tuners. If you're lucky, the '57 neck holes will be smaller than the Schallers, and you can just enlarge them to accommodate the new tuners. Otherwise you may need to find something else that will fit. Stewmac has a selection of tuners:
STEWMAC.COM : Bass tuners
I've swapped necks among Fenders several times without problems, by the way.