We have all three of his Tannahill recordings in the Whittaker Library - see here.
We also have his Robert Burns recordings - see here.
If you're interested in the old music publications that Fred referred to, we commend Geor Thomson's Scottish songbooks and R. A. Smith's Scotish Minstrel books, of which we have examples in our Special Collections. See our catalogue!
Another scholar who has worked on Tannahill's poetic contribution is Jim Ferguson, whose PhD at the University of Glasgow was a study of Tannahill's correspondence. You can read this online, here:-
Ferguson, Jim (2011) - A weaver in wartime: a biographical study and the letters of Paisley weaver-poet Robert Tannahill (1774-1810). PhD thesis.