Here are some short samples of music, played on the Cathedral organ, that could be used at weddings.

Louis Vierne: Finale from Symphony no. 1
Festive and exuberant, this staple of the organ repertoire makes a wonderful recessional piece.

J. S. Bach: Prelude in E flat, BWV 552
This is ideal as a solemn, dignified, entrance processional.

Jeremiah Clarke: Trumpet Voluntary
This well-known piece is equally appropriate at entrance or recessional.