I'm not sure why you say this. I am right now looking at the following paper of Ax
MR0229613 (37 #5187) Ax, James The elementary theory of finite fields. Ann. of Math. (2) 88 1968 239--271.
The first sentence is: "In this paper, we prove the decidability of the theory of finite fields and the theory of p-adic fields." He goes on in the introduction to explain exactly what this means: given a statement E in the language of rings, he gives an algorithm for determining the set of prime powers q such that E holds in the finite field of order q. He doesn't say anything about dependence on CH, and I don't see how this could possibly be the case ("decidability given CH" is a far cry from decidability!).
The only mention of CH in the introduction to the paper is in Theorem B, which states that a certain isomorphism of ultraproducts holds given CH. This makes more sense, since cardinality questions come into play in the isomorphism of elementarily equivalent objects.