Generation scheduling in future smart grids will face significant uncertainty due to their considerable reliance on intermittent renewable-based generation such as wind power. Adaptive robust optimization provides a suitable framework to handle wind-related uncertainty in generation scheduling. However, available robust models feature relevant practical limitations including 1) the potential lack of physical implementability stemming from disregarding the nonanticipativity of the dispatch process, 2) the potential suboptimality or even infeasibility due to the use of fixed-recourse schemes, and 3) the intractable computational burden associated with a scenario-based counterpart. This paper presents a new multistage robust unit commitment approach with non-fixed recourse relying on the formulation of an alternative two-stage robust model. As a result, the least-cost generation schedule ensuring dispatch nonanticipativity is attained by solving a trilevel program of similar complexity as compared with available formulations neglecting this aspect. Moreover, an enhanced column-and-constraint generation algorithm is devised whereby lexicographic optimization is applied to accelerate the finite convergence to optimality. Numerical simulations including a practical out-of-sample validation procedure reveal that the proposed approach is 1) computationally effective even for a benchmark that is well beyond the capability of a recently published method, and 2) superior in terms of solution quality over existing two-stage robust models disregarding dispatch nonanticipativity.