We study the roughening of a nonlinear elastic interface driven by temporally correlated noise. The nonlinear elasticity breaks statistical tilt symmetry. Using scaling arguments and a self-consistent effective-tension approach, we derive the crossover diagram and the steady-state structure factor. We identify three scaling regimes associated with the Larkin, anharmonic Larkin, and Edwards–Wilkinson universality classes, and obtain the crossover lengths separating them. Numerical simulations of large systems confirm the analytical predictions over the full parameter range. Our results provide a unified description of finite-size and crossover effects in a minimal nonlinear-elastic Ornstein–Uhlenbeck interface.