Swan upping dates, according to Stefan Buczacki's "Fauna Britannica", from around the Twelfth Century when swans were marked not for conservation but for quite the opposite, as a stock take of the Crown's preferred banquet delicacy. The Act of the Swan in 1482 extended consumption and ownership rights to landowners who were permitted to mark the birds to indicate ownership. Elizabeth I granted one exception, the swans at Abbotsbury; probably the largest muster of swans in the country.
The Act required the birds to be rounded up and marked, and sometimes pinioned, marking was by notches cut in their upper mandibles and numbers were recorded on Swan Rolls with disputes settled by Swan Motes. As Buczacki points out this was really a means of stopping the poor eating the birds - or more precisely exacting retribution on those that did!
The word "upping" is from the Sixteenth century and refers to the practice of upturning the birds into boats for the purposes of marking them. The national flock is known as the Queen's Game of Swans and the Queen "owns" all unmarked swans on open water. Landowner rights have dwindled so nowadays the only flocks upped under the terms of the Act asfar as I can discover, are those on certain stretches of the Thames which are owned by the two livery companies the Vintners and Dyers. The traditional (including fancy dress) Swan Upping on the Thames is done under the auspices of officials with titles such as the Queen's Swan Warden and the Queen's Swan Marker. The Crown's right of ownership continues despite the abolition of the Crown's rights to other animals in the early 1970s.
Needless to say there are many who regard the whole practice as archaic and cruel: birds can't fly away as they are mid-moult. At least the practice of marking the bill and clipping the wing feathers has been discontinued - though shockingly the former was only stopped as recently as 1998. The Queen's current Swan marker is David Barber and the current Swan Warden, Christopher Perrins