The one person I am struggling to find anything at all on so far for you is Lavinia, your grandmother

Can we discuss her dates and futher details?
This looks to be James at home still with parents on the 1881 census:
Alexander McLure 45 Traveller Serving Machiner b. Kirkmichael, Ayrshire
Hellen McLure 46 b. Gartsherrie
James McLure 20 coal miner b. Gartsherrie
Address: Colt Ter Clellands Buildings, Coatbridge, Old Monkland
Don't think Alexander had more children other than James and Sarah. Sarah, aged 8, shows in the household with parents and James in 1871 at North Square 210, Old Monkland.
And then 1891, I wondered if this was son James married to an Elizabeth. However, no sign of a Lavinia there

:
James McLure 30 coal miner b. Coatbridge
Elizabeth McLure 26 b. Shettleston, Lanarkshire
Sarah McLure 5
Helen McLure 3
Elizabeth McLure 1
Address: 11 High Sunnyside, Coatbridge
Monica