I may be mistaken but I believe F. W., Sternfeld, _Music in Shakespearean Tragedy_ (London/New York, 1963)_ remains the best place to begin looking for songs used in Shakespeare. The Willow Song occupies pages 24-52 in the book, with 5 or 6 versions including several with tabkature in facsimile. The Sternfeld book is alos a vaulable reference tool because he has tracked down original music for not only the tragedies, but the comedies and histories as well (see Index I).
Another volume which I have never seen was assembled y the late Andrew Charlton for Garland Press then in NYC> It is said to be a "practivcal" edition of the Sakespearean music. I tried to get it reprinted, but the Garland editor thought there would not be enough sales. But it can be found in libraries. The Sternfeld is also probably OOP, but it will be in many libraries. This should get you started. Perhaps Arnold Gessel will have something to add. ajn.