As soon as a wild animal crosses into somewhere you don't have permission to be in that is that.

As far as I am aware even if they succumb on that person's property the owner of that property then owns the carcass. Hence removing that carcass would be illegal.
And even worse if you ventured in to dispatch it, since they you are into armed tresspass. And firing into a property you don't have permission to shoot on would also be illegal, even if you were actively hunting the animal which had escaped into the adjacent property.

Could you not ring the landowner who owns the property you are on and ask them who owns the adjacent property?

Given that trespass is still a thing and the hunter would have to trespass in order to obtain the carcass, and the whole illegality of hunting on land that you don't have permission to be on, and armed trespass as you mention, and the part of the housing act 2002 which deals with bringing things onto the land which can affect the use of it by the owner, I cannot see a hunter doing this legally.

Also never heard of any cases of it, but yes they would be very interesting reading.

I'd be rather cautious of that site mind you, no sources that I can see which I would expect from something dealing with legalities.