Firstly, IANAL - this is just my findings from digging into this kind of scenario. Personally, I think blanket bans on open land like this are over reaching and unnecessary *provided* that the drone code is adhered to and the flight is safe.
However, there are some legal considerations here, even if you err on the side of caution and assume the byelaw actually exists (some landowners claim they have one when they do not; a byelaw has legal weight above and beyond normal landowner's rights and has to be applied for and granted). Many larger landowners do have such byelaws; many councils, The National Trust, and English Heritage for instance, and I would assume a park authority like the New Forest does too, but they are generally not specific to drones and may be being a bit free in their legal interpretation in order to include them in their granted byelaws. The NT's applicable byelaw dates from 1965 for instance, which obviously does not specifically cover drones, and actually states:
"No unauthorised person shall on Trust Property sell or offer or expose for sale any commodity or article or for the purpose of trade or reward take any photograph."
So, for commercial drone photography I'd say they're absolutely covered, but for purely recreational usage, especially given they don't have a blanket ban on photography and actively encourage it? Yeah, I don't rate their chances at all, but I still don't want to be the one that goes to court over it!
As far as the CAA is concerned they do not want to get involved in such cases and their default position is that provided there is no breach of the drone code, the take off/landing points are legal, and there are no safety incidents it is NOT their remit. Since they are responsible for *all* the airspace from the ground up in the UK (including Class G/uncontrolled airspace), the upshot of that is that if you can take off legally from public land, comply with the drone code, and do not have any safety incidents, then you are almost certainly free and clear as far as criminal law and the CAA is concerned.
Where you would have a possible legal problem is civil law. Byelaw or not, a landowner does have some rights to the airspace over their land, specifically "a right to to enjoy their land" and "a reasonable expectation of privacy". They are also perfectly entitled to issue a blanket ban on the operation of drones *from* their land, as is the case here (the drone code covers this too - you are supposed to have the permission of a landowner before you fly from their land). When it comes to an overflight, they do have the recourse of a civil suit against the pilot concerned - on either of the grounds above, or potentially a safety angle, including to farm animals and other wildlife.
If you do end up on the receiving end of a civil suit, then besides the obvious costs and other factors of going to court, the outcome will be very much down to the court depending on the specific scenario. For instance, there is no precise altitude defined where you would be legally OK for an overflight; whether you had breached the landowner's rights would be determined by the court based on any evidence presented.
On the plus side, come November (if all goes to plan), we'll all be registered, will have undertaken a mandatory safety exam of some kind, and will have some kind of card/registration number to prove it. Those default "In our experience, very few users have the correct training or permission from the Civil Aviation Authority to operate drones." style clauses go out of the window at that point as *all* registered drone operators will have the correct training and permission to operate from the CAA. That's going to make it much harder for them to justify their restrictions, and while I doubt they'll fold outright it *might* just result in some more latitude to fly because I doubt very much they want to risk a losing a test case and getting a precedent set.
