Abolishing private schools won't change anything other than there being one less place for wealthy parents to congregate at.
Abolishing it will mean now the state has to provide even more places as those same students will be needing a place, during the onset of the economic crisis many middle class parents were forced to pull their kids out of private schools and send them to state ones, that already created massive gridlocks all over the system.
As UK's schooling system today is very much a post-code lottery any home within a catchment area of a good school will simply be snapped up by the wealthier for their kids to go there, won't be long either before they make it difficult for less well off people to be there.
Now put this thought in your head, today the vast majority of the wealthier send their kids to be educated privately, in their private schools they will be among their peers, they shall be speaking about their latest pony, how daddy just bought mummy a brand new Range Rover and she is unhappy over the colour of the seats, then they shall be complaining that their home is so big they get lost in it, these same bunch will now attend your state schools and most likely they won't be mingling among you plebs but in their own clique of other students with wealthy parents because they aren't going to want to share their pony with you and most likely their mummy wouldn't want you to even touch her Range Rover let alone for you to get a ride after school in it.
Standards aren't going to improve by abolishing private schools, all that will happen is where it is deemed standards their kids are receiving isn't adequate they would simply hire the best tutor out there.... today I can easily hire an Oxbridge qualified teacher for as little as £30 per hour to tutor my kids and these tutors can most likely bring a barely passing kid to be an A* student within an academic year. I suppose these will go down in price if you get rid of private schools as there would be plenty of teachers, tutors and other academic staff that would be out of job, the rich will easily be able to afford them while you plebs will not so you will be back at square one.
What next would you like to abolish? Oxbridge?