Aren't you missing a variable for the cost the unskilled woman pays to have someone watch her children while she's being paid to watch the skilled woman's children?
Depends what level of income disparity you're operating on, rich people don't share nannies - the nanny's kids go to a nursery or more likely are looked after by a grandparent (maybe in a different country).
I can say this with some experience, rich people definitely hire nannies with children, and those children definitely often stay with the rich kids. Rich people get free playmate service for their child, nanny gets to raise their child in a nice environment and their child gets access to a wealthy social network.
It's even a common trope that shows up in literature fairly often.
You're ignoring the value the children bring to society by growing up and entering the labour force themselves. The highly skilled woman making a lot of money is unlikely to have her child make significantly more than her (regression to the mean).
On the other hand, empowering an impoverished woman to provide higher quality care for her children will significantly increase their upward mobility.
The highly skilled working mother can only spend less than (1 - marginal_tax_rate) * (1 - employer_profit_margin_on_labor) * $X on an hour of child care.
It was this exact calculation (and a high marginal rate) that led my own highly skilled wife to conclude that it wasn't worth working when we had our second child.
No, I'm fully aware of the fact that childcare today is often provided by people who are under-qualified for the task.
Which is one reason I think that basic income would broadly improve the level of care given to children. It would enable the people most qualified to give care--children's own parents--to provide that care instead of working excessive hours just to put food on the table.
In contrast, the unskilled woman who might otherwise work in day care only produces $X value/hour.
By taking the unskilled woman out of the labor force, you've destroyed $Y - $X of value.