Peter Scholze's [comment](https://mathoverflow.net/questions/427995/possible-characterisation-of-compactly-generated-weakly-hausdorff-spaces#comment1101190_427995)

>The correct characterization is that they are the topological spaces that can be written as filtered colimits of compact Hausdorff spaces along injective transition maps. It is pretty clear that all CGWH spaces are of this form (being CG, it is the filtered colimit of the images of maps from CH spaces; and those images are themselves CH by WH); the other direction is e.g. Proposition A.14 in Schwede's "[Global homotopy theory](https://arxiv.org/abs/1802.09382)".


 is the correct answer to my question: thank you for this!