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I used oneway.test() because Levene's test indicated heterogenity of variance. Unfortunately, I don't know how to calculate the effect size now. Does anybody have any ideas?

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I found a paper (EPS 625 – INTERMEDIATE STATISTICS ONE-WAY ANOVA – IN-CLASS EXAMPLE ADJUSTING FOR HETEROGENEITY OF VARIANCE, pdf) that provides the answer to this question however there is no reference to the equation mentioned, so one cannot be sure whether it is reliable or not. However, since it was produce by a University, it may be useful. $$ \text{estimated }\omega^2 = \frac{(df_{bet}(F-1))}{(df_{bet} (F-1)+N)} $$

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