It has been pointed out to me that Turkey and Mexico may not be appropriate comparators for the UK, so here is the chart published in the previous post, with 'new democracies' excluded.
The dispersion around the regression line is greater now so the R^2 is lower, but the broad trend remains. In particular, there are no cases in either the top right or bottom left corners: ie with very high inequality and very high female labour participation, or with very low inequality and very low female labour participation. If Willetts had a case, there surely would be.
Having said that, inequality and mobility are not exactly the same thing. A longitudinal analysis would be interesting, but cross-national work on this is diffcult, given the data limitations.