Evolutionary game theory has provided substantial contributions to explain the emergence of cooperation under unfavourable conditions in ecology, economics, and the social sciences. Recently, inspired by newly available empirical evidence on group interactions, higher-order networks have emerged as a natural framework to properly encode multiplayer games in structured populations. Here, we study the emergence of cooperation in a nonlinear public goods game (PGG) on hypergraphs, where collective reinforcement captures the synergistic or discounting effect associated with each additional cooperator. In well-mixed populations, single-order PGGs, where all games have the same number of players, display a change in the nature of transition from continuous to discontinuous depending on the exact form of nonlinearity. By contrast, mixed-order PGGs, where games with different number of players coexist, exhibit a richer dynamical regime wherein a state of active coexistence of bistability and cooperation can arise. We further find that scale-free hypergraphs promote cooperation, highlighting the crucial role played by both the initial placement of cooperators and the presence of hyperdegree correlations. Overall, our results provide a comprehensive characterization of nonlinear PGGs on hypergraphs and open up new avenues for richer models of evolutionary dynamics of multiplayer interactions on structured populations.