I think it's illegal to pick up some roadkill. You aren't actually allowed to pick up gamebirds (no puns please lol!) as, regardless of whether the gamekeeper cares, it is theft of property.
Hitting livestock (and dogs are still included in this clause) has to be reported - and it's sometimes in your interests because you can claim on the livestock owners insurance for damage to your car if the animal has escaped from a fenced enclosure (the owner is held responsible for allowing their livestock to stray).
And you can't take swans home cos they belong to her Maj' of course!