Folks, found a Bathgate, West Lothian marriage 1858, John Meek 27, Whitburn, master slater, son of John Meek, labourer & Jane Dixon, married Elizabeth Auld of Trees daughter of John Auld & Mary Shaw 17th October. If this is your connection they're not Irish at all, these Meek's have been in Bathgate since the 1500's.
Here's the 1935 will of a Janet French Walker/Meek of Blackfaulds, Fauldhouse, West Lothian, widow, died 15th February 1935 testate, £528 to Jeannie Wilson Meek, Elizabeth Meek & Mary Walker Meek all Blackfaulds.
Bests,
Skoosh.