http://www.informatik.uni-hamburg.de/~zierke/shirley.collins/songs/fairmaidofislington.html

Points out that probably the most common tune for it is "sellengers round", so you may find it helpful to search for that...

My former band used to do June Tabor's version, paired with "Under the Greenwood Tree". I've forgotten the chords, as it's quite a few years ago, but I don't remember it being at all complicated.