Two schools in my city converted from private to academy status five or six years ago. They basically converted because this city has an over-supply of private schools, and those two weren't keeping up with the competition - their results weren't as good, so they attracted lower-quality applicants, so their results went down further etc in a vicious circle. Their fees were also lower to try to draw people in, but that meant they couldn't pay to get the best staff, improve facilities etc. If they hadn't converted, I'm pretty sure at least one of them would have closed by now.
Since switching to academy status, they have both been massively oversubscribed with people wanting to get a private-school-style education for free (most other state schools here have a very poor reputation - results were amongst the lowest in the country for years). Obviously there were changes, like larger class sizes, but also benefits, like funding for expansion of buildings to cope with higher numbers, modernisation of facilities etc (this was a few years ago - this kind of funding may now have been cut, of course).
They are open to children from all over the city and the wider region, rather than a narrow geographic catchment area, and since they are hugely oversubscribed, entry to the academies is by banding: everyone takes a test to put them in an ability band, and the school picks a certain number from each band at random. This is meant to give a wide spread of abilities rather than cherry-picking the brightest.
However, it does mean that the only parents who apply are ones who are prepared to jump through the hoops of the testing system, and possibly to arrange transport for their children from a long way away, which of course is a form of selection in itself: having parents who are interested in and supportive of their child's education is a fairly crucial factor in getting good results.
The upshot has been that GCSE results of the first cohorts of pupils that have been through the schools from yr7 to yr11 are actually way better than they were when they were private schools. However, whether that would happen with the school you are talking about would probably depend on admissions procedures