The taking of road killed game, provided you did not kill it yourself, is unfortunately an urban myth. No game may be killed or taken in the UK without a valid game licence. All game is the property (live or dead) of the landowner whose land it happens to be on. The Deer Act 1991 states quite clearly that it is a criminal offence to enter any land (not your own) for the purpose of killing or the taking of deer (dead or alive). Often the game rights of adjoining landowers cover the whole of the road. In the case of there being different landowers each side of the road, then each landowners rights extend to the centre of the road. In this stituation, to pick up road kill from the road or verge would mean you are guilty of poaching.
Other roads are the property of local authorities and othe executive transport agencies. These roads, although, technically, in public ownership, do not give you any extra rights to pick up game. Such game is still not your property and you can be prosecuted under the game act, the 1994 Deer Act, the Theft Act 1968 (stealing by finding), as well as a possible conviction for poaching.
It is interesting to note that on summary conviction, the vehicle used to pick up such game will be confiscated by the the authorities.