I have a dc at state secondary, but friends with children at various state and private day & boarding schools.
The boarding school kids seem to be the least affected. Inevitably really, because, apart from holidays, they have built-in restrictions on movement and contact with people outside school, so they have ready-made bubbles. They also have lots of space, small class sizes and well-paid teachers who don't have to deal with most of the stresses that state school teachers face on a daily basis.
Other friends' children at private day schools seem to have a better set up for online schooling than my dc's state school, along with all the other advantages that help reduce the impact of Covid.
I know lots of children in state schools who don't have a quiet space to work at home, have slow internet and shared laptops. I don't know any child at private school who has missed the 11 days so far this year that my dc has, which seems about average for other children I know at state schools in my area. And online provision seems patchy in state schools, with little or no one-to-one teaching, so it's very hard for even the most motivated students, especially if they are struggling with topics.
Given that we have a government which seems hell bent on using the pandemic to make its rich friends even richer, I can't imagine they're losing much sleep about a greater attainment gap or have any plans to do anything about it. But sadly, where I live at any rate, many voters still seem to be swallowing the line that our PM is 'doing his best' 