In this paper we investigate the nonconforming $P_1$ finite element approximation to the sequential regularization method for unsteady Navier-Stokes equations. We provide error estimates for a full discretization scheme. Typically, conforming $P_1$ finite element methods lead to error bounds that depend inversely on the penalty parameter $\epsilon$. We obtain an $\epsilon$-uniform error bound by utilizing the {nonconforming $P_1$} finite element method in this paper. Numerical examples are given to verify theoretical results.