Hello Charlotte (sorry I don't remember receiving a notification of your post) and Tony. I do not have a Wallace in my database (yet) but it is far from complete and unfortunately I am not familiar with the area to get a sense of the different villages but the proximity is promising!)
Do you have any more information on who the parents of your great grandfather Peter Halliday were? So far, in the mining areas in North Lanarksire (Carluke, Cambusnethan, Old and New Monklands, Shotts, Dalziel, Bothwell....) I have found that many of the Hallidays descend from either Robert Halliday (d. 1851) and Margaret Williamson or Archibald Halliday (1786-1840) and Marion Penman. I believe that Robert and Archibald, if not actual brothers, were closely related.(I have found a hypothetical link : if they are brothers, their parents may have been Thomas Halliday and Ann Campbell of Perth... I have no confirmation of this however) If you do find a connection to either of these two families, I will happily share what I have. Cheers.