The Coronation of the King and Queen Consort takes place in Westminster Abbey. Though a large church, Westminster Abbey is not a cathedral. It is an abbey church, and a Royal Peculiar.
But what does this mean?
A Royal Peculiar is a jurisdiction which is not under the authority of the diocesan bishop, but rather under the oversight of the Crown. Normally parish churches and cathedrals are under the jurisdiction of the diocesan bishop: for instance, in London St Paul’s Cathedral relates to the Bishop of London. While a full part of the Church of England, Westminster Abbey is under the jurisdiction of the sovereign. The Bishop of London is also the Dean of the Chapels Royal, so has a role in the chapels in the royal palaces.
Other Royal Peculiars include the Chapel Royal, St James’s Palace, the King’s Chapel of the Savoy, and St George’s Chapel, Windsor among others. Peculiars owe their status, in many cases, to their royal connections or foundation, or to other factors which meant that they have been historically separate from the diocesan structure of the Church.