Ah, the good old Roadkill/Poaching myths.
Deer. If they are wild, then anyone can pick up the carcase even if you have run it over UNLESS it has run from a deer farm or a Crown Estate. Wild animals belong to no-one until rendered into possession by killing them.
Deer are NOT classed as Game under the Game laws.
Wild game (Partridge, pheasant, grouse etc) does not belong to anyone, so if you find it on the road you can keep it, according to the game laws. You cannot sell it ten days after the start of close season however.
Rabbits are not game. They are wild, do not belong to anyone, and so you can pick them up and keep them. They only become ground game for the purposes of certain land issues, none of those are relevant to roadkill. On a road they are simply wild animals.
Now for the difficult bit.
The Wildlife and Countryside Act makes it all a bit more complicated. It is an offence under that act to be in possession of any wild bird except for a game bird, and then only in the close season for that game bird. So, if you decide to pick up a dead pheasant form the side of the road you CAN , but only between October 1st and February 1st.
There is also an enormous schedule of wild animals that you may not have in your possession, or any part of,or any derivative, dead or alive. The exceptions are similar to the ones above re game, for species of deer outside of their closed seasons, so you need to know when they are.....
I'll stop now as my brain is addled. It all gets very convoluted, but I think the Highways Agency eventually decided as they have the responsibility to clear the roadkill off the road, it actually belongs to them. Anyone who can research all the various bits of game laws, common law, the theft act, and bring it all together given the vagueries of the Wildlife and Countryside Act to create anything approaching a rule of thumb, should be knighted or decorated or whatever.
In the meantime, nowhere in any of the legislation is there anything to say which driver can stop and pick it up. That is a pub-barrister's red herring, a rural myth (as opposed to an urban one....somehow urban roadkill doesn't have the same appeal!) As most of the game laws were tidied up and conglomerated in the 1800's I somehow doubt if the lawmakers were thinking about killing game with cars!
So. Can you pick up roadkill? Yes, but then again it all depends.....
Can anyone remember reading any local court reports or local paper headlines about people being prosecuted for picking up and eating roadkill? No, didn't think so! SO the likelihood of prosecution is minimal.
If you were faced with a tasty morsel on the tarmac would you take it home and feed it to your family or fledglings?