They would probably be buried at Coldingham Priory unless they had some connection with another village or parish. Borders FHS and the Friends of Coldingham Priory recorded the Monumental Inscriptions last year so if they have a memorial, we should have the inscription. If they don't have a memorial, the next step would be to contact Bereavment Services at Scottish Borders Council who may have a record of the burial.
This is a problem I do not know whether she was buried Catherine Docherty/doherty or Douglas, The child was named James Docherty or Douglas on the death certificate. She was catholic but the family disowned her and I think she may have been buried in a paupers grave.