I do not know if there is actually an English Woodcock, so your knowing you don't know is actually more knowledge than I truly have on the subject. ;o) I'll put that as one in your column.
And I'll refrain from sharing what I do know about what we just call Woodcocks around here... except to dispel any notions of impropriety by saying they are birds and make a fair meal for one. Hm. It brings to mind a book written in 1963, American printing but with a great deal of English (and European) history concerning guns and their employ: The Passion of the Gun. Good bed side topical reader. Not much about wood cocks, but you'd maybe be surprised how much fowling went on in your fair land not 100 years ago.