We first establish strong convergence rates for multiscale systems driven by $α$-stable processes, with analyses constructed in two distinct scaling regimes. When addressing weak convergence rates of this system, we derive four averaged equations with respect to four scaling regimes. Notably, under sufficient Hölder regularity conditions on the time-dependent drifts of slow process, the strong convergence orders are related to the known optimal strong convergence order $1-\frac{1}α$, and the weak convergence orders are 1. Our primary approach involves employing nonlocal Poisson equations to construct ``corrector equations" that effectively eliminate inhomogeneous terms.