In this note it is shown how to some extent the Edgeworth expansion for the distribution function of Kendall's $\tau$ can be established by using a well-known general result on such expansions.
"A Note on the Edgeworth Expansion for the Kendall Rank Correlation Coefficient." Ann. Statist. 6 (4) 923 - 925, July, 1978. https://doi.org/10.1214/aos/1176344267