A web search suggests that the category of CGWH spaces was introduced in the paper "Classifying Spaces and Infinite Symmetric Products" by M. C. McCord (Transactions of the American Mathematical Society Vol. 146, (1969), pp. 273-298).
McCord motivated introduction of his "weak Hausdorff" separation axiom by noting that
"the requirement of the Hausdorff condition can be a problem because certain standard operations on spaces can lead outside the category", in particular quotient spaces in algebraic topology and topological algebra.