Actually, the functions referred to in the question don't make any use of their inputs at all. They could just as well have been defined as functions from the e

Actually, the functions referred to in the question don't make any use of their inputs at all. They could just as well have been defined as functions from the empty set to {0, 1}. The "f" in the question is not a function, it's a label, such that the referent of f if God exists is the function f1, that always outputs 1, and the referent of f if God does not exist is the function f0, that always outputs 0. The question is actually not about computability at all, it's about labels.