Alexander and Elizabeth were married (or at least the banns were read) in Riccarton on 11 May 1804. (source IGI)
Unlikely that document, available on SP for a small charge, will show you much detail but might possibly have fathers names, ie JW's grandfather.
Depends on how much you wish to pursue their line to try and find a connection to yours.
There are a few possible births for Alexander on IGI and SP but nothing definite from the few searches I've done.
Have you thought about checking MI's for the area. Having money, it is quite possible JW had a stone erected for his parents / family?
Dave