We report high-pressure x-ray diffraction, x-ray absorption spectroscopy, and electrical transport measurements on CuMO2 (M=Cr, Mn) delafossitelike oxides in an attempt to study their structural and electronic evolution with pressure. Recent studies of the similar CuFeO2 delafossite has revealed a pressure-induced breaking of the unusual high axial anisotropy resulting in a structural phase transition coinciding with the metal-metal intervalence charge-transfer phenomenon. The present study revealed other possible scenarios responsible for the collapse of the high axial anisotropy and evolution of the O-Cu-O bonds in delafossitelike materials under pressure. Thus in CuMnO2, the O-Cu-O dumbbells tilt with respect to the c axis at P>13 GPa, but in contrast to CuFeO2, the tilting is continuous with pressure increase, justifying a second-order phase transition within the C2/m structure. Meanwhile in CuCrO2 (R¯3m) the first-order structural phase transition to the monoclinic structure (P21/m) is observed at about 26 GPa characterized by the discontinuous bending of the O-Cu-O bond in contrast to the tilting in the case of CuFeO2 and CuMnO2. In both studied systems, we did not find clear evidence of valence transformations, similar to that observed in CuFeO2.