I am wondering about the effect of a werewolf bite on a magical being.
The best canonical evidence I've found is the following (emphasis mine):
Werewolves spend most of their time as humans (whether wizard or Muggle). Once a month, however, they transform into savage, four-legged beasts of murderous intent and no human conscience.
Once a month, at the full moon, the otherwise sane and normal wizard or Muggle afflicted transforms into a murderous beast.
-- Fantastic Beasts and Where to Find Them
Now this could just be common wizarding prejudice, forgetting about the non-human sentient beings such as house-elves, goblins, and centaurs. But my interpretation is that only humans can be turned into werewolves. The effect of a werewolf bite on a house-elf or goblin might just be the same as the effect of any other animal's bite, or at worst the same as the effect of a werewolf's bite on a human while the werewolf is in human form (e.g. Bill Weasley).
From Pottermore - Werewolves there are a few pieces of information pertinent to the question.
While human, the werewolf may be as good or kind as the next person. Alternatively, they may be dangerous even while human
Notice it explicitly states that whilst a werewolf is human.
where werewolves have married human partners
Again human partners, nothing mentioning any other species.
The most interesting quote I found talks about the differences between a werewolf attack on a wizard and one on a muggle.
In the late nineteenth century the great English authority on werewolves, Professor Marlowe Forfang, undertook the first comprehensive study of their habits. He found that nearly all those he managed to study and question had been wizards before being bitten. He also learned from the werewolves that Muggles ‘taste’ different to wizards and that they are much more likely to die of their wounds, whereas witches and wizards survive to become werewolves.
So now we know that Muggles are unlikely to survive an attack from a werewolf and also that nearly all werewolves were wizards. You could interpret this as there being other species, but the more likely route is that it is simply referring to muggle werewolves.