Hi Jamie
With the Irish side for John, it can be hard to trace back particularly when you are not sure where in Ireland John was from

I tried searching on Scotlands People for a death for mother Catherine with the names Burke and Connolly but no joy (nearest are two entries with a birth year of 1840 +/- 5yrs - too late to be John's mother).
There are a couple of trees on Ancestry for the family. From John and Margaret's Catholic marriage, may explain why there is nothing at all also showing regarding Ireland as official registration did not begin in Ireland until 1864. Just on the outside chance, I don't know how practical this is for you. The marriage shows at St Patrick's Chapel in Anderston. Apart from the official registration of the marriage, have you ever thought to follow up with the church's archives for the original entry of the marriage. In Roman Catholic marriage, details of where someone is born could be recorded by the church. Might be something to follow up on.
What does Charles' marriage cert to Christina McKay say about his father. Does it show him as deceased at the time of his marriage and what occupation?
Monica