It occurs to me that the death certificate of Mary Bishop or Nimmo could be very useful. At the moment, the latest sighting of her seems to be that 1851 census. Has anyone found her in 1861 or later? Or a death certificate? Though there doesn't seem to be one in the SP index.
(I don't think she can be the Mary Bishop who died in Whitburn in 1871, aged 82. There is a Mary Bishop, widow, aged 69 at Half Way House, Whitburn in the 1861 census. In 1841 there is a family at Half Way House consisting of John Bishop, 55; Mary Bishop, 45; Sarrah Bishop, 15; John Bishop, 10; Sarrah Millar, 10; Sarrah Millar, 75; and two servants. So I think this Mary Bishop is probably Mary Millar, who married John Bishop and had a family in Whitburn including Sarah, baptised 30 August 1821, and John, baptised 11 December 1829.)
Could Mary Bishop or Nimmo have emigrated with her son William, or perhaps joined him later?
Also intrigued by Janet D Brash, who is listed in 1841 with the Bishop/Nimmo family as a baby and then as Mary Bishop or Nimmo's granddaughter in 1851. Who is her mother? Could she be the Janet Dick Brash who married John Calder in Uphall in 1877? The 1841 says she was one month old, which fits with the 1851 saying she was 9. So she was born in April/May 1841, and therefore conceived in July/August 1840. She can't be the child of Mary Nimmo, who was only 9 in 1841. Could she be the child of Margret Lowe, aged 21? And could Margret Lowe be a daughter of Mary Bishop, born before Mary married John Nimmo .... Eureka! There is a baptism of Margaret Low in Livingston on 16 January 1820, parents Lawrince Low and Mary Bishop. So if I am right, John Nimmo was Mary's Bishop's second husband. Janet's marriage certificate might clarify that.