  In this paper we study the influence of electron screening on the
Bohr-Weisskopf (BW) effect in many-electron atoms. The BW effect gives the
finite-nucleus magnetization contribution to the hyperfine structure.
Relativistic atomic many-body calculations are performed for s and p_1/2 states
of several systems of interest for studies of atomic parity violation and
time-reversal-violating electric dipole moments -- Rb, Cs, Fr, Ba+, Ra+, and
Tl. For s states, electron screening effects are small, and the relative BW
correction for hydrogenlike ions and neutral atoms is approximately the same.
We relate the ground-state BW effect in H-like ions, which may be cleanly
extracted from experiments, to the BW effect in s and p_1/2 states of neutral
and near neutral atoms through an electronic screening factor. This allows the
BW effect extracted from measurements with H-like ions to be used, with
screening factors, in atomic calculations without recourse to modelled nuclear
structure input. It opens the way for unprecedented accuracy in accounting for
the BW effect in heavy atoms. The efficacy of this approach is demonstrated
using available experimental data for H-like and neutral Tl-203 and Tl-205.
