No problem.

There are a few places like this where we have normal cycle route signage, but which go across private land. Although it's unlikely the landowners would just revoke access, they could lock the gates without needing a traffic order and users wouldn't have any recourse.

At the other end of The Greenway, the road works on High Street have a TTRO (temporary traffic regulation order), but as far as I can tell the related closure on The Greenway itself doesn't.