Oh, that's terrible.
Hopefully, when you tell the idiot what has happened, they will offer some recompense. And think twice another time.
I don't know, but I'd think that if you own your land then they'd need permission to come onto your property, but the permission may be implied unless you state that you expressly do *not* give them permission. Which, given what's happened, you may well decide to do. (And then you'd need to check whether any pheasant shot by them but falling onto your land were legally yours..
)
What they can or cannot do if they have reason to suspect that a fallen bird is not dead, I am not sure - but it sounds as though there was someone at home in this case, so all they needed to do was knock on the door and ask...
What a horrible start to the New Year for you.
I hope that's your bad luck for the year done and dusted, and the rest of the year will be a good one.