All swans are owned by the crown and the monarch has the exclusive right to kill and eat them.
Or at least that's the way I heard it, come to think of it I have no idea at all if that's true. Stops people killing and eating swans though. Not that many would anyway these days.
St John's College serves swan on formal occasions sometimes, because they have some connection to the royal family that means they have special permission. (Or used to in the Queen Elizabeth days, I don't know if they still do under Charles)
Not all, just mute swans; so not Bewick’s or whoopers (if I remember correctly)
Edit:
“His Majesty specifically owns any unclaimed mute swan in open water in both England and Wales in a ceremonial fashion. This has been a law since medieval times. His ownership is shared with the Worshipful Company of Dyers, granted to them by the Crown in the 1400s.”
Or at least that's the way I heard it, come to think of it I have no idea at all if that's true. Stops people killing and eating swans though. Not that many would anyway these days.