  We derive novel, fast, and parameter-robust preconditioned iterative methods
for steady and time-dependent Navier--Stokes control problems. Our approach may
be applied to time-dependent problems which are discretized using backward
Euler or Crank--Nicolson, and is also a valuable candidate for Stokes control
problems discretized using Crank--Nicolson. The key ingredients of the solver
are a saddle-point type approximation for the linear systems, an inner
iteration for the $(1,1)$-block accelerated by a preconditioner for
convection--diffusion control, and an approximation to the Schur complement
based on a potent commutator argument applied to an appropriate block matrix. A
range of numerical experiments validate the effectiveness of our new approach.
