Gee, I never thought I'd ever have to look Adele up on Wikipedia, which says that she was born in Tottenham (North London), moved to Brighton (south coast) at 9, then to Brixton (South London) at age 11 and, soon after, to South Norwood (also South London).
So, she's a Londoner for sure but not truly a Cockney, as this term traditionally refers to the East End of London and just about edges its way across the River Thames to Bermondsey. In my youth, it was said that a true Cockney was born within the sound of Bow Bells (the bells of St Mary-le-Bow church), though this definition was applied in a more geographical sense than a literal sense.
Incidentally, Dobells, the rather well-known jazz, blues and folk record shop (no longer with us), used to advertise that the tru jazz fan was born within the sound of Dobells.
Culturally, the term Cockney refers to people from working class districts of London not just in the geographical areas of London that I've indicated above but also in some of the adjoining areas.
Linguistically, the Cockney accent, patter and speech patterns (especially the fast-talking, quick-witted and fairly chipper style) has spread even further.
As a West Londoner and even as someone from a working class background in West London, I would never be considered a Cockney by anyone from other parts of London, especially by a true Cockney. That said, Brits from other parts of Britain often consider anyone from the London area (and sometimes anyone from the South-East of England more broadly) to be a Cockney.
Finally, I did once buy a couple of Adele CDs but they were presents for someone else.