Influence of the plasma collisions on the laser-driven collisionless shock formation and subsequent ion acceleration is studied on the basis of two different collisional algorithms and their implementations in two well-known particle-in-cell codes EPOCH and SMILEI. In this setup, an ultra-intense incident laser pulse generates hot-electrons in a thick target, launching an electrostatic shock at the laser-plasma interface while also pushing the interface through the hole-boring effect. We observe, to varying degrees, the weakening of the space-charge effects due to collisions and improvements ($ge 10%$) in the energy spectra of quasi-monoenergetic ions in both PIC codes EPOCH and SMILEI. These results establish the `collisionlessness of the collisionless shocks in laboratory astrophysics experiments.