Short answer: This is open for the specific “topologically slice but not smoothly slice knot” construction. We do not currently know whether two different knots K and K′ of that type can produce nondiffeomorphic exotic R^4’s via the standard construction. It is conceivable (and consistent with present knowledge) that all such R^4_K could even be diffeomorphic.

Context and references:

• The construction you have in mind (originating with Akbulut) takes a topologically slice knot K and produces a smooth 4–manifold R^4_K homeomorphic to R^4; if K is not smoothly slice then R^4_K is exotic. This is a standard way to produce a “small” exotic R^4 (i.e., one that embeds in S^4).
• While there are known methods to produce infinitely many and even uncountably many pairwise nondiffeomorphic exotic R^4’s (e.g., via Casson handles, radial families, end-sums, etc., due to work of Gompf, Bizaca–Gompf, Taubes, and others), distinguishing explicit small exotic R^4’s produced by a given ad hoc recipe remains hard. In particular, no known invariant is effective in telling apart the R^4_K that arise directly from varying K in this particular construction.
• This situation is discussed in the literature; see, for example, R. Gompf and A. Stipsicz, 4-Manifolds and Kirby Calculus (Cambridge, 1999), Section 9.4 on small exotic R^4’s, where it is noted that for certain explicit constructions (including those stemming from knots) it is unknown whether varying the input changes the diffeomorphism type. More recent overviews reiterate that distinguishing small exotic R^4’s is still subtle and that there is no available tool that would separate the knot-based examples.

Related remarks:

• If one allows different constructions (e.g., using genus-at-infinity invariants derived from embeddings into closed symplectic manifolds and adjunction inequalities), one can produce infinite families of pairwise nondiffeomorphic small exotic R^4’s. However, these do not establish that the particular family {R^4_K} coming from topologically slice, non-slice knots contains more than one diffeomorphism type.
• There are many families of topologically slice, non-slice knots (e.g., positive untwisted Whitehead doubles of positive knots) detectable by τ or s, but at present those invariants do not translate into known invariants of the resulting open 4–manifolds that would distinguish the smooth structures R^4_K.

Conclusion: With current techniques, it is unknown whether two different topologically slice, non-slice knots can give rise (by this specific construction) to two nondiffeomorphic exotic R^4’s.