Data package accompanying the publication with the above title in the Journal of Chemical Physics [10.1063/5.0117867]. This package provides all the relevant information for the numerical study of the crystal nucleation of highly-screened charged colloids. We strived to answer the question: when can nearly-hard colloids be seen as hard spheres? To this end, we compared the nucleation barriers and rates of nearly-hard colloids to that of hard spheres by mapping via an effective hard-sphere diameter, which is a common practice in experimental studies. In brief, this package contains the data required to reproduce the graphs, as well as the simulation source code and analysis scripts required to process the raw output. In each directory there are README.txt files that describe the content. The main directory contains three directories: the directory [Figures], which contains the published graphs, and the directories [HS] and [HCYukawa], which contain everything on the systems of, respectively, hard spheres and hard-core Yukawa particles. In these last two directories you will find the directories [Data], [Mathematica], and/or [Code]. The directory [Data] contains all the processed data (.dat format) for that system. Each subdirectory in [Data] contains its own C codes (and bash scripts) used for generating the raw simulation data. The directory [Mathematica] contains all the Mathematica notebooks used for processing the raw data and visualizing the results. Lastly, the directory [Code], when present, contains additional C codes used for analyzing the raw data.