Collinson says that Katherine Beardie is the name affixed to an air in the Rowallan MS as well as the Skene MS, and Chambers [1842] tells us more about these.
There's a 1883 arrangment by Charles D'Esteve of a quadrille of children's songs in the Mitchell Library that includes the earliest 4/4 version of the 6/8 Sherrifmuir tune I could find, with the title Broken Bridges. J J Fuld says the first appearance of London Bridge with the modern tune was printed in 1879 in Philadelphia.   
I give transcriptions of various versions of these tunes in my book.