Fruitveg and Skoosh.
Some more information for you.
As you say, Alexander Miller Meek had six children.
William 1867-1939, married Marion (Minnie) More, and had four children themselves. William did carry on the boot manufacturing business. He joined the Linlithgow Town Council, became a magistrate, and then Provost of Linlithgow. His son William carried on the business.
Jane Bird (1870-1951) married Andrew Grierson, and had three children. Andrew became town clerk of Edinburgh and was knighted, so Jane became Lady Grierson.
John (1872-1952), married Isabella Moffatt, and with his brother Alexander and Walter Bellis founded Bellis and Meek. I believe they had two children.
Mary Miller (1876-1958) did not marry. She lived at Rosland Cottage and then Edinburgh.
Alice (1878-1955) married Archibald Blackadder, motor engineer, They did not have any children. Alice died at Rosland Cottage.
Alexander (1883-?), as you know, married Edith Montgomery. I am aware, I think, of three children, John, Rhona and Ethel, but have no further details.
I hope this helps to fill some gaps.
Bill66