We present some theoretical predictions concerning the amplitudes and magnetic sensitivities of the linear polarization signals produced by scattering processes in some UV and FUV spectral lines of the upper chromosphere and transition region, such as Ly-alpha and Mg II k. To this end, we have calculated the atomic level polarization (population imbalances and quantum coherences) induced by anisotropic radiation pumping in semi-empirical and hydrodynamical models of the solar atmosphere, taking into account radiative transfer and the Hanle effect caused by the presence of organized and random magnetic fields. The amplitudes of the emergent linear polarization signals are found to vary typically between a fraction of a percent and a few percent, depending on the scattering geometry and the strength and orientation of the magnetic field. The results shown here encourage the development of UV polarimeters for sounding rockets and space telescopes with the aim of opening up a true diagnostic window for magnetic field measurements in the upper chromosphere and transition region of the Sun.