It's early for Irish records, albeit some do exist, though often not online.
It can be helpful if you know religion.
Have a play around with the surname tool at
https://www.johngrenham.com/surnames/especially adding in the second surname.
This will give you an idea of the spread of the Bamford (and variants) surname around mid 1800s, and an indication of where both surnames appeared in same area. Although a bit later than the timescale for your couple, it should give you some areas to consider a bit more closely.
Also, have a look at the instances of a John Bamford in the Tithe Applotment records. Again, whilst it may not include your John, it can give an indication of possible areas with the surname present in 1820s and/or 1830s.
What respective parents' names have been recorded on the death registrations of John and his wife (assuming they died in Scotland after mid 1850s that is)?
For the son, Thomas, in Scotland:
Name Thomas Bamford
Sex. Male
Father's Name John Bamford
Father's Sex Male
Mother's Name Helen Currie
Mother's Sex Female
Event Type Christening
Event Date 26 Sep 1841
Event Place Kilsyth, Stirlingshire, Scotland, United Kingdom
so, from checking on Scotlandspeople, it is a Church of Scotland record.