The crystal structure and superconducting properties of a new type of titanium-pnictide superconductor, BaTi2(Sb1-yBiy)2O (y = 0.2, 0.5, and 0.8), are comprehensively investigated over a wide pressure range to elucidate the effect of substituting Bi for Sb on the superconducting behavior. The behavior of superconducting properties under pressure changes drastically with y, as expected from the double-dome Tc-y phase diagram obtained at ambient pressure. In this study, three BaTi2(Sb1-yBiy)2O samples (y = 0.2, 0.5, and 0.8) are considered, which correspond to the first superconducting dome, nonsuperconducting part, and second superconducting dome, respectively, in the Tc-y phase diagram. The crystal of BaTi2(Sb1-yBiy)2O with y = 0.2 shows a clear collapse transition, i.e., a drastic shrinkage of the lattice constant c at ca. 5 GPa. Strictly speaking, the collapsed crystal phase coexists with the noncollapsed phase above 5 GPa. On the other hand, BaTi2(Sb1-yBiy)2O with y = 0.8 shows a continuous change in the crystal lattice with pressure, i.e., no collapse transitions. The pressure dependence of Tc for BaTi2(Sb1-yBiy)2O with y = 0.2 shows a drastic increase in Tc at approximately 5 GPa, where the collapse transition occurs, indicating a clear pressure-induced superconducting phase transition related to the collapse transition. The value of Tc for BaTi2(Sb1-yBiy)2O with y = 0.8 increases slightly up to ∼2 GPa and is almost constant at 2-13 GPa. It is found that the superconducting behavior under pressure can be unambiguously classified by y based on the double-dome Tc-y phase diagram, indicative of distinguishable superconducting features at different y values. In this study, we comprehensively discuss the superconducting properties of the exotic material, BaTi2(Sb1-yBiy)2O, with a double-dome Tc-y phase diagram.