Hi Alison,
Can I ask how you know that John Burnside/Nelly Miller have a known connection to John Burnside/Sarah Allen if it's not known how they are connected?

Have you looked at the 1911 Irish Census for Killyberry [http://www.census.nationalarchives.ie/pages/1911/Londonderry/Bellaghy/Killyberry/607722/ ]? There is a Robert Burnside with wife Lizzie and a brood of children including
Kennedy Burnside. With the family are Robert's brother John [34] and his sisters Jane [43] and Lizzie [31]. Might be a coincidence!
Also in Killyberry in 1911 are another Robert Burnside with wife Lizzie and his mother Sarah, who I believe to be Sarah Allen who incidentally is my great great grandmother

It's possible that John Burnside [husband of Nelly Miller] was the brother of [or otherwise related to] the William Burnside Princess has mentioned but I haven't been able to get back beyond the marriage of John and Sarah. I also have a copy of a will relating to the death of John Burnside in 1908, who I believed to be the husband of Sarah - I don't have it to hand, so I'll look it out later in the week.
Hopefully the 1901 Irish census will give us all some answers next week!
Best wishes
Rockford