The River Bourne flows from the Vale of Pewsey down through Salisbury Plain to Laverstock. It must join the Avon somewhere near the B & Q on the Southampton Road, but I don’t know if the place where the rivers join is visible.
Riverbourne Road is near the river, in Milford.
I believe the area where the two rivers joined was once called Mumford. The joining of the river is indeed underground and I think the river was rerouted during all the building.
Ian is right:
According to http://www.wiltshire.gov.uk/community/getcom.php?id=136
“There has been continuous settlement here from the 6th century, and probably much earlier, with 6th century pottery being found at Milford. Apart from Laverstock there was another Saxon settlement at Mumford, probably at the confluence of the rivers Avon and Bourne, which survived until later medieval times.”
Thanks,
Matt