The finite cell method, and more broadly, unfitted finite element methods have been gaining popularity as alternatives to classical finite element methods for the solution of partial differential equations as the need for a boundary-conforming mesh, a typically manual and laborious task, is circumvented. The resultant system from such discretization methods are, however, numerically ill-conditioned; consequently, specialized iterative solvers are necessary for the efficient solution of large-scale finite cell systems. We focus on the finite cell discretization of the Stokes and the Navier-Stokes equations in this work and present an adaptive geometric multigrid (GMG) solver for the solution of the model problems. We present two Vanka-type smoothers for the treatment of the cutcells in the finite cell method and discuss their performance and computational cost. The convergence of the GMG method, both as a standalone solver and as a preconditioner within Krylov accelerators, for linear and nonlinear cases is discussed. It is shown that the iteration count of the solver remains bounded independent of problem size and the depth of the grid hierarchy. Finally, it is demonstrated that the GMG method can be used for the efficient solution of large-scale linear and nonlinear finite cell flow problems.