We present a systematic delay-adaptive prediction- based control design for nonlinear systems with unknown long actuator delay. Our approach is based on the representation of the constant actuator delay as a transport Partial Differential Equation (PDE) in which the convective speed is inversely proportional to the unknown delay. We study two different frameworks, assuming first that the actuator state is measured and relaxing afterward. For the full-state feedback case, we prove global asymptotic convergence of the proposed adaptive controller while, in the second case, replacing the actuator state by its adaptive estimate, we prove local regulation. The relevance of the obtained results are illustrated by simulations of a biological activator/repressor system.