From an interview:
Director Brian Percival reveals that in 90 per cent of the scenes featuring animals, they actually used prosthetics. "We were not allowed to put an animal through something that it didn’t need or that we weren’t trained for," he says. "Hence the prosthetics, but the animals we did work with were so well trained. We had a bull, a huge animal the size of a small horse, and we wanted it to look angry, but it was so good natured we had a few problems."
Season 1, Episode 3 centers heavily on the death of a racehorse, wherein the animal is put down for its own health.
Although conversation surrounding the death spans nearly the entire episode, the scene in question starts at 25:30 and ends around 26:59. During that time, you can hear an off-screen gunshot at 25:55 (which is heavily implied to have been the instrument responsible for euthanizing the horse). Additionally, you can catch glimpses of the body from 26:26 to 26:34, and once again from 26:46 to 26:59. There is no blood or gore---for all intents and purposes, it looks like the horse is just lying down---but the implication may still be disturbing for some.
Lastly, there's 1 scene (split in half) where a vet goes to conduct an autopsy on the racehorse. The first half starts at 40:52 and ends at 41:35 before cutting away. The show then cuts back to the autopsy at 44:24 and concludes at 44:49. Again, no blood or gore is seen: in fact, the horse is hardly shown at all---just a bunch of close-ups on the characters.
SPOILERS! In season 2 episode 1, Tristan takes a budgie out of its cage and the shock of being picked up is fatal. There are shots of the bird lying still at this point and later when the bird’s body is put into a drawer.
Season two episode two: surgery on a cow's stomach is clearly shown for over a minute. And there are several animals who need to have things 'burst' in the first two episodes for season two which is often quite graphic.
Well, three or four people get scratches either from a car or from falling off a horse. And a cow is cut open to find cause of death and the man who does it cuts out the heart and then shows it. So it's not excessively gory for a farm vet show, in my subjective opinion.
Season 2, episode 6: a dog is hit by a car and taken to the vets. By the time he is examined he has died.