I think there is a misunderstanding underlying your question. The fact that the material is a superconductor does not imply that electrons are flowing at relativistic speeds.
1) In a superconductor the current is carried by the condensate, not by random motion of electrons. In that sense, there is no drift velocity. The formula for the current density is
$$
\vec\jmath = \frac{e n_s}{2m}\hbar\vec\nabla\phi
$$
where $n_s$ is the superfluid density, and $\phi$ is the phase of the condensate wave function. Note that
$$
v_s = \frac{\hbar}{2m}\vec{\nabla}\phi
$$
is the velocity of the supercurrent.
2) At $T=0$ we have $n_s\to n$, and the first formula is essentially the same as the usual formula for the current density, $\vec\jmath=en\vec{v}$, so a naive estimate of the drift velocity from the known current and density of electrons give the correct supercurrent velocity (even though the physical mechanism is quite different).
3) The current density in a superconductor is not larger than what can be achieved in ordinary conductors (in fact, it is often smaller). Superconductors have a critical current (roughly, the kinetic energy of the supercurrent cannot exceed the condensation energy). As a result the supercurrent velocity is limited to speeds much smaller than $c$.