Differing opinions about Tyburn River

An e-mailer (Simon Crome) has made the following remarks about an entry in the UCLE section of this site:
     In the item on Rivers in London used as Literary Allusions, it is 
     said that the Tyburn flows in a pipe through Sloane Square 
     underground station.  This is a mistake - the river in the pipe 
     is the Westbourne, which rises near Westbourne Grove, flows 
     through the Serpentine, then goes underground to Sloane Square 
     (through which it is conveyed in that pipe) and finally emerges to 
     flow into the Thames at the eastern end of the Royal Hospital

In July 2002, another e-mailer (Diana Clements, a geologist) disagrees in part:

     Just to correct you on the source of the Westbourne:  It rises from
     springs in the Hampstead region where the permeable Bagshot Sands
     and slightly-less permeable Claygate Beds on the top of the Hampstead
     Heath overlie the impermeable London Clay.  Springs gush forth from
     both these junctions to feed the Westbourne, Tyburn, Fleet, Brent and
     others on the north side.  The Westbourne then flows down through the
     Westbourne Grove area.  It probably got its name from being on the 
     'west' of the 'bourne'.  Nicholas Barton in his 'Lost Rivers of London'
     gives a good description with a map detailing the route.  You are quite
     right about Sloane Square!