This one is rather obscure and I am not that confident that it is the answer but "Children Children" written by Paul McCartney features the line "I know a tyne waterfall."
Yes it is. "Ferry Cross the Mersey" was recorded by Gerry & The Pacemakers and I think was written by the group leader Gerry Marsden. This one possibly appears to be just the name of a river within one of the song lyrics rather than accompanied by the word "River" itself. I shall ponder some more . . .
Following my previous suggestion @ 17:38 on 07/07/17 this next reference may also be very tenuous . . .
According to some sources the Beatles' song "Get Back" began as a satirical and critical look at attitudes towards immigrants in Britain and it was loosely based on British politician Enoch Powell's notorious 'Rivers of Blood' speech.
If River Wye from "Ask Me Why" is considered plausible, then how about River Wear from the lyrics of several songs - eg. Come Together has the lyric "He wear no shoeshine...".