This paper introduces the Heterogeneous-Fleet Electric Vehicle Routing Problem with Nonlinear Charging Functions (HEVRP-NL). This problem involves routing a heterogeneous fleet of electric vehicles, utilizing multiple charging modes, and accounting for time-dependent waiting time functions at charging stations. The problem is modeled using a path-based mixed-integer linear programming formulation. To solve this problem, we present an algorithmic framework that alternates between two components. The first component is an iterated local search algorithm with a problem-specific route evaluation function, which obtains local optimal solutions and generates a pool of high-quality routes. The second component is a set-partitioning model that combines a subset of routes from the pool into a feasible solution. We design HEVRP-NL benchmark instances based on the publicly available electric fleet size and mix vehicle routing problem instances, which are used to evaluate our methods. For small-scale HEVRP-NL instances, the proposed model can be employed in a general-purpose mixed integer programming solver to achieve optimal solutions or find good upper bounds. This exact approach serves as an evaluation of our heuristic algorithm's ability to attain optimal solutions rapidly. Extensive computational results on large-scale HEVRP-NL instances illustrate the advantages of considering non-linear charging functions and show the impact of waiting time at the charging stations. Finally, we conduct experiments on 120 benchmark instances for the E-VRP-NL and 168 benchmark instances for the E-FSMFTW-PR, which are the special cases of our problem. The results indicate that our algorithm outperforms existing approaches from the literature and identifies 32 new best solutions for the E-VRP-NL and 33 new best solutions for the E-FSMFTW-PR, respectively.