Try Kenneth McKellar (great thread on Mudcat in the past, with links to many YouTube clips) -- he toured extensively and had a great repertoire of songs to appeal to expats, including Burns, Harry Lauder, and music hall.

Other good songs that would have featured in the kind of social sing-around at Hogmanay that preceded TV were 'The Rowan Tree', 'The Road to the Isles', 'A Wee Deoch an' Doris'; 'The Skye Boat Song', 'Mairi's Wedding', 'Bonnie Mary o' Argyle' and -- of course -- 'Loch Lomond'.