John McAllister, mmn Lithgow, died in 1913 aged 67. So it looks as if he was born 1845/1846.
His marriage certificate says he was 23, which suggests 1846/1847.
In 1871 John, 23, and Elizabeth, 22, are in Glasgow with their son John, under 1 year and cousin Sarah McKee, 26. John was born in Ireland, which would account for not being able to find a birth/baptism record in Scotland.
In 1861 there's a household consisting of James Law and his wife Margaret, both born Ireland, two children born in Glasgow, and a 14-year-old son, John McAllister, born Ireland. I infer from this that Francis must have died quite young and Margaret Lithgow then married James Law.