I have heard that deer belong to the crown and that it is questionable whether you are ever legally permitted to pick them up, but it is not something I am sure about and my personal feeling is that police have better things to do than chase down people picking up road kill. However, I am certain that if you are the driver responsible for having hit an animal, you are not meant to pick it up yourself. This is to prevent people using it as an excuse for mowing down wildlife for fun and profit.
From all other aspects, the deer that you have hit yourself is going to be the least likely to cause you problems. A colleague of mine hit a fallow a few years ago and mentioned it to me the following day. He did not think the deer was badly hurt since he thought it ran off. I went to investigate just to practice tracking. In fact, the deer was about 2 metres from the road in a clump of trees and had clearly died very shortly after impact. Since I knew when it was hit, that the driver hadnt called any vets and that the carcass was invisible until you were next to it, I did not have to worry about a vet having shot it full of barbiturates. The intestines had got punctured, which wouldnt have mattered much if I had found it the night before, but a day of lying on its side had tainted all the haunch on the downhill side. The other haunch, the shoulders and the backstrap were all fine. I didnt even try to gut the animal, just took what I could without puncturing anything unfortunate. The meat didnt taste particularly strong.