Dr James Currie (1756 — 1805), Burns's first editor and major biographer writes in his The Works of Rober Burns:

"The ch and gh have always the gutteral sound. The sound of the English dipthong 00, is commonly spelled ou. the French u, which is often heard in the Scottish language, is marked oo or ui. ...."

This advice is repeated in various other books on Burns. The English long oo used by trained singers was to make the songs more suitable for the drawing room, sung round the pianoforte. It's like a breath of fresh air hearing Burns sung by 'folkies'.