A quantitative analysis is developed for the effect of external pressure on critical temperatures in Hg(N)- and Tl 2 (N)-superconductors with N = 1,2, and 3 CuO 2 layers per molecular unit, and with chemical composition HgBa 2 Ca N - 1 Cu N O 2 N + 2 + δ and Tl 2 Ba 2 Ca N - 1 Cu N O 2 N + 4 + δ , respectively. As in earlier work, Cooper pair formation is assumed to arise from indirect exchange pairing between conduction electrons via closed-shell oxygen anions (O 2 - ), in a s-wave BCS formalism. The exchange coupling is obtained as a function of three parameters α, β, and κ, with α and β (Gaussian) parameters in the wavefunction for the oxygen valence band and the conduction electrons, respectively, and κ = bk F α 1 2 , withb a materials constant for the cuprates; k F is the length of the Fermi vector. Comparison is also made with experiments in which, for the Tl 2 series, pressure is applied (or changed) either just above T c or at room temperature. Agreement with experimental results is quantitative throughout; in particular, the high T c values of 154 K (for Hg(N = 2)) and 160 K (Hg(N = 3)) are quantitatively reproduced, at the pressures observed. They are due to two-dimensional characteristics of superconductivity in these systems. The anomalous behavior of T c (P) in the Tl 2 (N) compounds with pressure applied at ambient temperature is found to result from pressure-induced diffusion of oxygen anions from the (Tl-O) 2 bilayers towards the CuO 2 layers. Neither van Hove singularities nor pressure-induced changes of the hole concentration in the CuO 2 layers are found to play a role in superconductivity of these compounds.