Superradiant instabilities of spinning black holes (BHs) can be used to impose strong constraints on ultralight bosons, thus turning BHs into effective particle detectors. However, very little is known about the development of the instability and whether its nonlinear time evolution accords to the linear intuition. For the first time, we attack this problem by studying the impact of gravitational-wave (GW) emission and gas accretion on the evolution of the instability. Our quasi-adiabatic, fully-relativistic analysis shows that: (i) GW emission does not have a significant effect on the evolution of the BH, (ii) accretion plays an important role, and (iii) although the mass of the scalar cloud developed through superradiance can be a sizeable fraction of the BH mass, its energy-density is very low and backreaction is negligible. Thus, massive BHs are well described by the Kerr geometry even if they develop bosonic clouds through superradiance. Using Monte Carlo methods and very conservative assumptions, we provide strong support to the validity of the linearized analysis and to the bounds of previous studies.