In a Banach space, we consider the nonlinear Langevin time-delay differential equations with ψ-Caputo fractional derivatives. Using weighted norms involving Mittag-Leffler functions, we obtain some existence and uniqueness of solutions of the problem. Besides, in some cases, the condition for the problem having a unique solution can be relaxed using Burton's method. We also obtain some Ulam-Hyers and Ulam-Hyers-Rassias Mittag-Leffler stability results for the main equation. Two examples are given to illustrate our theoretical findings.