Under
Section 4.4 of the Theft Act 1968, when an animal is killed on the roads it is "reduced into possession" and becomes the property of the landowner. To pick it up or otherwise remove it would therefore constitute theft.
"
Wild creatures, tamed or untamed, shall be regarded as property; but a person cannot steal a wild creature not tamed nor ordinarily kept in captivity, or the carcase of any such creature, unless either it has been reduced into possession by or on behalf of another person and possession of it has not since been lost or abandoned, or another person is in course of reducing it into possession."
As has been said, it would have to be a pretty mean-spirited judge to convict someone of this. I wonder if it's ever happened??
The idea, by the way, was to prevent people specifically going out with the intent to knock down and kill wild animals. At least that's how I understand it to be.