I found some of the following information in a book called Reminiscences of an Orkney Parish. The recipe was gleaned from family sources. It would appear the young gentleman in the song is rather more partial to his liquid refreshment than the lasses.

Bride's Cog

On the wedding day the guests assembled at the bride's house at the appointed hour, and were ushered into the barn, which was seated with planks round the walls. At the door the people were welcomed by the bride's father, or by some highly respected elderly relative or neighbour, who for the time received the title of "Master-Hoosal) (Master of the household).When all were seated this dignitary handed the Ale Cog to the most honoured guest, who, having taken a deep draught of the liquor passed it to his neighbour, and so it went round the company, following, of course, the Sun's path in the heavens. The Cog was a circular vessel, made of wood, formed by staves secured by wooden hoops and having two, or sometimes three long upright handles, called "Horns" rising from its brim. Those tub-like drinking vessels were of various sizes and different styles, and held from One Quart Imperial two pints Scotch some had every alternate stave made of dark wood, and instead of "Horns" two elaborately carved "Lugs" or handles. This kind of Cog had a more artistic appearance than the "Horned Cogs", but even these latter were very neatly finished, and there "Horns" were nicely "Turned".

Traditional Cog Mixture

6 Bottles of home brew (home made beer)
4 Bottles of Wee Heavy Ale
2 Bottles of Whiskey
½ Bottle of Port
½ Bottle of Rum
Demerar'a Sugar
If a milder taste was required the Rum was changed for Brandy.

Method
Put beer in suitable container, heat till warm enough not to burn the drinker, then add the spirit and sugar to taste, and serve.