I think their idea is that the state would simply take over the schools, the buildings, the staff, and the endowments, thus making them comprehensives. (This is a bit of a legal minefield, though).
In practise, this would mean, same school, but whoopee, no fees to pay.
(not me, my DC went to state schools)
Alternatively, they could abolish the charitable status, but then there would be no incentive for schools to maintain their bursaries, or share their facilities, their various outreach programs.
Also, I think this is a bit of a minefield, as not all independent schools have charitable status, and it depends on the legal definition of a charity, and how the school is setup.
However, supposing all independent schools were somehow abolished, then I guess the smaller schools would close, forcing those students into the state sector. The more prestigious schools would just move abroad, to France or Ireland, and remember, many have overseas students which currently add to the economy, would thus benefit other countries.
Would this include Jewish Schools, Madrassas?
And where would Diane Abbott and Shami Chakrabarti send their children to, once independent school are abolished?
Assuming you managed to abolish all independent schools, rich people would continue to have extra private tuition for their children, as I believe, Tony Blair did for one of his.
You could try to abolish private tuition, but how would this be monitored? Would it include ballet lessons, football coaching, music lessons, or just after school extra maths lessons?
Once you have abolished all extra private tuition, then remember that teachers are parents too, and you would have to ban them from teaching their own children, or taking them to museums, etc.
As many dedicated parents give their children such an unfair advantage, you could try and create a society where children are removed from their parents as much as possible (to make it fairer).
Some people think we are getting there already, as nowadays both parents often have to work, and children placed in day care.