We prove Gaussian concentration inequalities for maximal displacement of compactly supported random walks on a compactly generated locally compact group with polynomial growth. Concentration inequalities with different exponents hold for non-centred random walks as well, after correction by the drift. When the support of the measure generates a virtually nilpotent group, we provide an effective version of this result. These more refined estimates rely on the existence of a ``quantitative splitting'' of a virtually simply connected nilpotent group, a result which may be of independent interest. As applications, we deduce that the same concentration inequalities hold for centred random walks on the following classes of groups: amenable connected Lie groups (including non-unimodular ones), polycyclic and more generally finitely generated solvable groups with finite Prüfer rank. This shows in particular that centred random walk are diffusive on such groups. For polycyclic groups, this strengthens and completes partial results previously obtained by Russ Thompson in 2011.