PL locally flat knots have rank 2 $PL(2)$ normal bundles, but $PL(2)$ can be reduced to $SO(2)$ structure group. So there is a PL embedding of the smooth manifold $\nu S^2 \cong S^2 \times D^2 \hookrightarrow S^4$. This induces a smooth structure on a neighbourhood of the knot, namely the image of $\nu S^2$. Since the embedding is PL, there is a PL structure on the exterior of the knot $S^4 \setminus \nu S^2$, and the PL structures on the intersection $S^1 \times S^2$ coming from the exterior and coming from the smooth structure on $\nu S^2$ agree. Then use: given a PL structure on a 4-manifold with boundary $X$, together with a smooth structure on $\partial X$ refining the PL structure, there is (up to isotopy) a unique smooth structure on $X$ refining the PL structure (and restricting to the given smooth structure on $\partial X$, but this is automatic by uniqueness of smooth structures on 3-manifolds). This extends the smooth structure on $\nu S^2$ to all of $S^4$. Since this smooth structure on $S^4$ refines the standard PL structure on $S^4$, it is isotopic to the standard smooth structure on $S^4$. This isotopes the knot to a smooth embedding in the standard smooth structure.
Partial attempt at references. PL bundles: Rourke-Sanderson, Kuiper-Lashof. PL $\Rightarrow$ smooth for 4-manifolds: the key is Cerf's result $\Gamma_4=0$. Main reference is Hirsch-Mazur, Smoothings of Piecewise Linear Manifolds,