We report both zinc-blende (ZB) and wurtzite (WZ) crystal phase self-assembled GaAs quantum dots (QDs) embedding in a single GaAs/AlGaAs core-shell nanowires (NWs). Optical transitions and single-photon characteristics of both kinds of QDs have been investigated by measuring photoluminescence (PL) and time-resolved PL spectra upon application of hydrostatic pressure. We find that the ZB QDs are of direct band gap transition with short recombination lifetime (~1 ns) and higher pressure coefficient (75-100 meV/GPa). On the contrary, the WZ QDs undergo a direct-to-pseudodirect bandgap transition as a result of quantum confinement effect, with remarkably longer exciton lifetime (4.5-74.5 ns) and smaller pressure coefficient (28-53 meV/GPa). These fundamentally physical properties are further examined by performing state-of-the-art atomistic pseudopotential calculations.