I don't reckon they will.
They said they would back in the 90s, as I remember my parents talking about it, and had plenty of time to consider it with their "education education education" and never did.
And the people it will drive out from private schools will be disproportionally the people who aren't well off but have sacrificed on other things to do it, especially people who's dc for whatever reason, often SEN, aren't coping in mainstream.
The real rich people will hardly notice.
So they'll be adding more children, to our already full and underfunded schools. That's really going to help the state system, isn't it? They're probably hoping a number will just home school them so they can forget about them.
They'll also have to fund however many more school places in places which are already over full, putting more strain on schools especially subjects where teachers are already in short supply etc.
Round here the schools are full or pretty much full all the way up except a few - the ones the parents don't want. Most schools don't have the space to expand for an extra class either. Fewer parents will get their first choice school. More children won't be taught by a qualified teacher.
They'd be better to bring back assisted places in greater numbers.
No, I don't have children at private school, so no skin in the game.