We study Abelian Berry phases in the almost-bosonic anyon gas, which is described by a two-parameter Chern-Simons-Schrödinger energy functional. We show that the low-energy states of the model are well approximated by states of the nonlinear Landau levels (NLL). The NLL is a family of explicit soliton solutions (states) parametrised by coprime and complex polynomials. Using these explicit states, we derive expressions for the Berry connection and compute Berry phases associated with adiabatic transport along closed loops in the parameter space given by vortex positions, scaling, and interaction terms. The Berry phase can be interpreted as the holonomy of the U(1)-connection. Our results indicate a dependence of the Berry phase on the topology of the underlying configuration space. Our work contributes to the understanding of geometric aspects of many-anyon systems and conceptually connects to holonomic and topological quantum computation.