Hi there,
Boo, and welcome to the Downs! The book and recording you remember is called 'The Road Goes Ever On'; Swann composed the melodies to Tolkien's poetry, and in one case, Tolkien himself sang him the melody as he imagined it, and Swann wrote it down. I own the book with the notes and have enjoyed playing them; except for "Errantry", which has a more difficult piano accompaniment, they're not hard to play for an average pianist. I also have the recording; the singer is William Elvin (what an appropriate name, eh?!).
Tolkien and his wife Edith (who was herself an accomplished pianist) heard and enjoyed at least one performance of these songs - at their golden wedding anniversary, if I remember correctly.
More information can be found here:
Wikipedia entry
Donald Swann's website
Amazon
By the way, it's always nice to have new members who are old-timers to Tolkien! If you want to get to know the Downs' geezers, do look into
The Tolkien Coming of Age Club, where we share our experiences - or just companionable silence, whichever we enjoy most at the moment.