I think you'll find it's the opposite way round to the way you think, in that Fyffe wrote it and tried to get Lauder to record it, but he was already a star and didn't bite on it, maybe due to the boozy nature of it all. So Fyffe had the recording. According to Wiki the first recording was 1920.. (it may have been written in that year, but I can only see the first performance use in 1921 newspapers. From Songfacts " Will Fyffe. According to Richard Anthony Baker in British Music Hall: An Illustrated History, this was one of two songs Fyffe wrote for Harry Lauder while touring in revue, but Lauder rejected them both."   Anyway, it's public domain now.