The title says it all.
This wisdom of crowds more or less prefers wood pigeon. A Google search finds a million results for “wood pigeon” and a quarter of that for “woodpigeon”, though several of the top results have both spellings.
The BTO and RSPB prefer “woodpigeon” and ebirds goes for a third alternative - “common wood-pigeon”. Google gives more results for “wood-pigeon” than for the other 2; while I treat Google’s counts of results with a degree of skepticism, I find this particular outcome not credible. (I suspect some vagary in how Google handles hyphens.)
Anyway, I would have said that either spelling was acceptable. I consider “woodpidgeon” and “wood pidgeon” as archaic.
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And I was inspired to take a look at Google Ngrams
Hyphens are un-fashionable 
!
there are 28 occurrences of Wood Pidgeon in iSpot (3 are mine)
628 Woodpigeon including this https://www.ispotnature.org/communities/uk-and-ireland/view/observation/876083/ (ID Title)
256 as Common Wood Pigeon and
140 as Wood Pigeon
All three are in the common-name menu
Google Ngram viewer was new to me. Thanks. I think I am going to like it.