Your question assumes the wire to be a perfect electrical conductor (PEC). This basically means that its conductivity goes to infinity $\sigma \rightarrow \infty$, or equivalently its resistivity goes to zero. Recall from Ohm's law that:
$$\mathbf J (\mathbf x ,t) = \sigma \mathbf E (\mathbf x ,t)$$
Where $\mathbf J$ is the volume charge density. Now for a PEC, as $\sigma \rightarrow \infty$ , you can see that if we had a non-zero electric field inside the conductor, the current density would also go to infinity, which is clearly nonphysical. Thus, the only way to satisfy the above equation is for $\mathbf E (\mathbf x ,t)$ to be zero everywhere inside the conductor, at all times .
Thus, for a perfect electrical conductor, the electric field is zero inside the conductor even in time-varying cases.
Note that this is of course an idealization, and any wire has some finite conductivity, and thus the electric field inside is not exactly zero, although it is very small (I am disregarding things like superconductors). For a quantitative measure of how good the conductor rejects the internal electric field at some frequency , one can use the penetration depth, which is the the length you need to go inside the conductor for the field to become $1/e$ times its surface value. For further information about penetration depth check this answer.
Also note that the fact that your wires carry a charge doesn't affect the above reasoning, which means that the electric field is also zero in your specific problem.