I hope these work, as me and music and technology is not a good mix. Music is for vinyl (or CDs...)!
Ralph Vaughan Williams
The Lark Ascending not for any one character, but for The Shire.
More Vaughan Williams
Fantasia on a Theme by Thomas Tallis which I think fits with Bilbo's leaving of The Shire.
Link to page with Vaughan Williams samples
Holst's
Mars would go with any battle, but especially with the armies of Sauron marching for Helm's Deep. And Holst's
Jupiter suits Tom Bombadil as Jupiter is the "bringer of jollity".
Link to page with Holst samples
Mozart's
Dies Irae from Requiem is diabolical so I think might suit Saruman.
Link to page with Mozart samples
And Elgar's
Nimrod would go with a solemn moment such as the fall of Theoden in battle - probably influenced by hearing it at so many Remembrance Sunday ceremonies.
Link to page with Elgar samples