Sybil Reader wrote a book on Whitburn, there is a section in there on William Wilkinson and the house. It's called Olde House now, A wealthy businessman built a house on the bank opposite and took the name of Whitburn House.
As for any further assistance with the tree, I haven't got a clue. Took a look, realised that there were endless possibilities and no census and gave up.
If you find anything of interest I would like to know myself.