Greenapple,
The Patersons may have been poor but, at least they could afford to buy a plot, unlike many other working-class people of the time.
There are only 2 internments in this plot between the time the cemetery opened (1861) and when my records end at 1915. There seems to be a mix-up in the parents names, either by yourself or the original informant.
JOHN PATERSON. Coal miner, Chapel, aged 39 years, married.
Parents - William Paterson and Margaret Bulloch.
Interred in Private Cemetery No.65. 20th March 1866.
JANE PATERSON. Housewife, Morningside, aged 50 years, widow.
Parents - Walter Bulloch and Margaret Gilchrist.
Interred in Private Cemetery No.65. 28th December 1876.
Chapel and Morningside are in the Newmains area of Cambusnethan parish.
I couldn't find any other reference to this family in the cemetery records, did they have children?
There are a few Bullochs in the Newmains and Wishaw areas, Walter Bulloch married Marion Robb at Cambusnethan in 1857. They had a plot in an area of the old churchyard that I am pretty sure is part of a Victorian extention of the burial ground.
The name Walter, along with his abode, makes me think that there may be a connection.
Walter and Marion had 2 children who died in infancy,
ELIZABETH, aged 14 months, interred 6th Sept 1867 and INFANT SON, aged 14 days, interred 15th June 1863.
These 2 were paupers funerals but both would have been buried in the family plot.
Other records for this plot -
owned by Walter Bulloch and Marion Robb.
MARION BULLOCH, aged 49 years, interred 11th December 1888.
WALTER BULLOCH, aged 63 years, interred 28th July 1898.
There are 5 other entries, all children, surnames Bulloch, Eaton, Clark and Kennedy, all from the 1890s.
Monumental Inscription taken from South Dalziel churchyard in Motherwell.
"Erected by Walter Bulloch in memory of his wife Christina Waddell who died 14th July 1893 aged 54 years. His father William Bulloch died 24th November 1887 aged 87 years. His mother Janet Anderson died 14th November 1865 aged 42 years". This may or may not be the same family but it's worth taking note of it.
Good luck,
Lodger