The Place of Execution derives its name from the Tyburn Brook, not the River Tyburn.
The Tyburn Brook is a tributary of the River Westbourne and is unrelated to the River Tyburn. The River Tyburn splits beneath St. James Park with one branch going southeast toward Westminster and the other heading southwest to join the Thames near Vauxhall Bridge.
The Tyburn Brook does not join the River Fleet at any point, joining the Thames beneath Blackfriars Bridge.