It has recently been discovered that a fraction of ultraluminous X-ray sources (ULXs) exhibit X-ray pulsations, and are therefore powered by super-Eddington accretion on to magnetized neutron stars (NSs). For typical ULX mass accretion rates (1019 g s-1), the inner parts of the accretion disc are expected to be in the supercritical regime, meaning that some material is lost in a wind launched from the disc surface, while the rest forms an optically thick envelope around the NS as it follows magnetic field lines from the inner disc radius to the magnetic poles of the star. The envelope hides the central object from a distant observer and defines key observational properties of ULX pulsars: their energy spectrum, polarization, and timing features. The optical thickness of the envelope is affected by the mass losses from the disc. We calculate the mass-loss rate due to the wind in ULX pulsars, accounting for the NS magnetic field strength and advection processes in the disc. We argue that detection of strong outflows from ULX pulsars can be considered evidence of a relatively weak dipole component of the NS magnetic field. We estimate the influence of mass losses on the optical thickness of the envelope and analyse how the envelope affects broad-band aperiodic variability in ULXs. We show that brightness fluctuations at high Fourier frequencies can be strongly suppressed by multiple scatterings in the envelope and that the strength of suppression is determined by the mass accretion rate and geometrical size of the magnetosphere.