The school has never been selective so the only criterion was catchment. The could have moved in temporarily with relatives or rented something like this, but only for application. When I was told about that, the child was definitely commuting from their family home in another borough, the family never sold it. Of course it's shit but shit happens all the time.
Camden, Islington, Haringey, Hackney to some extent are home to champagne socialists, Labour politicians, journalists etc so of course they work the system all the time. Especially these days when going private is very problematic if you want an ideologically based career.
Tory have more integrity sometimes - Cummings is also a local and he was pestered by paparazzi but he didn't move his children to a donor's penthouse (his are younger of course but one could argue they could have been upset too).