I did do some research on this a while back and then as usual got sidetracked into something completely different.
But Janet is right, Atte just means at or near, and so John Atte Wood, just means John who lived near the wood. Subsequently searching through websites like British History Online, it is quite a common name and trying to connect them just based on a common surname is bound to be incorrect. Also although some of them might have developed into the surname Atwood or variations,others might have become Wood, or something else entirely.
I think in terms of naming in Wikitree, I would continue to follow the Euroaristo/medieval guidelines and use Wood as the Last name at birth, and Atte Wood as the current last name. Then Atwood or whatever could be in other last names?