I wouldn't bet the house on this as there are no test cases I know of, but......
I believe its legal prior to shooting a deer but I presume you mean once it is shot, you have wounded the deer and it needs following up. Yes it is legal under the hunting act. If your stalking for.....
preventing or reducing serious damage which the wild mammal would otherwise cause—
to livestock,
to game birds or wild birds (within the meaning of section 27 of the Wildlife and Countryside Act 1981 (c. 69)),
to food for livestock,
to crops (including vegetables and fruit),
to growing timber,
to fisheries,
to other property, or
to the biological diversity of an area (within the meaning of the United Nations Environmental Programme Convention on Biological Diversity of 1992),
obtaining meat to be used for human or animal consumption, or
participation in a field trial.
You must use only 1 or 2 dogs and be on the land legally. There are other conditions, but thats the jist of it
However if you have a wounded deer and it needs dispatching, say the the road side after an RTA or other justified reason. If using a .22lr or a knife is the best way, your justified under various laws/act etc that also applies to using a dog, if you think its the best way, even if you have just shot it.