You're right that effect size statistics for nonparametric tests usually can't be expressed in terms of variance explained.
Spearman correlation is equivalent to Pearson correlation on the ranks of the data. I suppose you could convert this, by squaring rho, to "variance of the ranks of y explained by the ranks of x", but I suspect relying the audience to understand rho is a better approach.
For Wilcoxon-Mann-Whitney, an easily understandable effect size statistic is Vargha and Delaney's A, which is simply the probability of an observation in one group being larger than an observation from the other group. This could also be used for pairs of groups in Kruskal-Wallis.
There are other effect size statistics used for nonparametric tests that can be difficult to interpret in words. For example, there is an epsilon-squared used for Kruskal-Wallis. It's a useful statistic, but not easy to interpret in words.
There is also an "eta-squared" used for Kruskal-Wallis, which is supposed to be more related to the usual eta-squared for an anova on the ranks of the y variable. (eta-squared is the same as the r-squared from an anova). But I'm not that enamoured with this statistic.
Likewise there is an "r" commonly used for Wilcoxon-Mann-Whitney, but it doesn't have great properties for an effect size statistic. It's better to use Vargha and Delaney's A, Cliff's delta, or rank biserial correlation.