I had four books with Greensleeves/What Child is This, and there were four different chord arrangements. I question whether there is a single definitive chord sequence.

It seems to me that I read somewhere that "What Child is This" was based on or derived from "Greensleeves," implying that it's not necessarily exactly the same tune. Sounds the same to me, although I haven't rigorously compared them note for note.

Here is, in my opinion, an elegant chord sequence for "What Child is This." It has fewer chord changes than most arrangements, and I think it's the better for it:

       Am                 G            
  What child is this, who laid to rest,
     Am            Em                  
  On Mary's lap is sleeping,           
       Am                G             
  Whom angels greet with anthems sweet,
        Am        E7        Am         
  While shepherds watch are keeping?   
  C             G          Em          
  This, this is Christ the King,       
       Am                  Em          
  Whom shepherds guard and angels sing.
  C               G         Em         
  Haste, haste to bring Him laud,      
      Am        E7     Am              
  The Babe, the Son of Mary.           

At the end of the final (3rd) verse, on the last word--"Mary"--use A rather than Am. I understand this is called the "Piccadilly third," and comes from an era where it was thought unseemly to end a song on a minor chord. Whatever the reason, I like the effect it has on the ending.