There is a lot of detailed information on this website. The number of dogs isn't important, but if your dog accidentally chases a wild quarry species you are not breaking the law.
In practice however, hare coursing is endemic in some parts of the country, and a lot of working sighthounds are still used for illegal quarry. High profile prosecutions of people like the Heythrop are a distraction and probably only help to generate donations.
I doubt any landowner is going to worry about your dog bringing down a deer - roe and muntjack in particular are real pests and can do serious damage to woodland. The more pressing question is what are you going to do if your dog brings down a deer? I wouldn't fancy trying to put even a muntjack out of it's misery tbh. Do you have the stomach to slit it's throat?
A cautionary tail for you - I was out walking my two greyhounds and two jrt's (one mine, one a friend's) and they put a young roe up. My younger grey ran it down in the middle of a ploughed field. The other three piled in and although I would prefer that it had a clean death, I didn't have a knife or any other means to kill it cleanly so I left it for the dogs to do it for me. The deer was crippled and would have died a long, slow death anyway. Of course I ran into the game keeper on the way back - I had deer blood on me from the dogs, and he was very blase, asked if it was dead and went to find the carcass.
My other big worry with the greyhounds and deer/hare was always if they chased them onto a road, I don't know if that may be a consideration in your case?
One of our dogs raced off after a muntjack... Luckily I was with DH who stayed out to look out for him. I went to wait at the village pond in case he got bored and wandered back home. He did. Up the main road through the village, with a traffic jam behind him . I talked to some other dog walkers while I waited. Apparently most farmers/ keepers won't miss a deer much, but the deer have a nasty kick and could easily win. It seems they have a fearsome reputation...
MuttDog was chased by a Muntjack . RottDog had sent it into the woods and it went crashing into MuttDog, it was obviously pissed off and butted her before giving chase. Mutt came crashing out of the undergrowth at top speed with her tail firmly between her legs.
I would. Have the stomach to slit it's throat tbh if I had dh's knives in the car. And I'd probably take it home for the pot.
Her recall isn't too bad, but once a terrier sees something tasty looking, they get in the zone, don't they? It's for that reason I'd never have her off lead near livestock. (although she's fine with horses.)