I wanted to let you settle in before discussing wildlife, but the conversation has come up, so...
I've lived in various degrees of rural most of my life, and I have two dogs with very high prey drive: they are absolutely bred to hunt, and do so under control. Gaining that control is bloody difficult (especially when your spouse is crap at dog training), and takes time and dedication, and in our case e-collars (which I barely need to touch around prey now: both dogs will either not chase at all, or stop on the whistle).
So I understand where you are coming from, having had a terrier who would fanny off rabbiting, and having watched one of my current dogs bolt across a road in pursuit of a hare (it was that made me e-collar train her: hares were her crack cocaine and nothing else had worked).
Leaving aside the law and the risk to the deer, chasing deer can be very dangerous for dogs. I know one couple whose previous dog was killed by a deer (kicked by a red deer) and know of another (friends of friends) whose dog was killed by a muntjac. They can also wind up getting lost, getting onto roads, or chasing livestock. A friend's dog chased a deer onto a road. Fortunately the lorry barrelling along was able to stop in time, and the driver (IIRC) caught the dog. Traffic was stopped in both directions on a busy trunk road until she arrived, panting, to collect him.
Besides the risk to the dog, I know a lot of landowners who wouldn't be happy with someone else's dog (or even their own) chasing deer through their woods and fields. Controlled culling is one thing; uncontrolled pursuit is something else.
I'm not having a go, honestly. I agree that dogs need to run (especially the high-drive breeds), and I know how hard it can be to keep them under control. Just please be aware of the risks that you are running. You have the most gorgeous dog who you obviously love very much, and given his breeding he needs to run - but he needs to be under your control when he does so.