What to do if your dog's nail starts bleeding because it was cut too short at the groomers:
1. Keep calm. It can be stressful to deal with a quicked nail, but the calmer you are, the less distressed your pet will be. A lightly quicked nail is uncomfortable - briefly - for your pet, but as long as you don't get too upset, your pet will soon forget it ever happened.
2. If you're still nearby, bring the dog back! Groomers and vets use styptic powder or nitrate sticks to quickly and easily stop a nail bleed.
3. If you have styptic powder at home (you can get it at your local pet supply shop or online: https://amzn.to/2I07AK0), it's easy to apply. Just blot excess blood with a paper towel, wet your fingertip and press it into the styptic powder to pick up a small amount, then press the powder against the cut edge of the nail and hold. You can hold for a full 3 minutes to be certain the blood has clotted, but the powder usually works in 10 - 20 seconds. Check that the bleeding has stopped. If not, repeat.
4. If you don't have styptic powder, you can use one of the following powders instead: sugar, flour, corn starch, or baking soda. You'll probably have to keep pressure on for at least a minute or two, depending on the severity of the bleeding. It helps greatly if you can keep your pet nice and calm (anxiety raises their heart rate and makes them bleed more). You can also hold ice wrapped in a towel against the foot to constrict the blood vessels and slow the bleeding.
5. You can use peroxide to clean up any blood stains on your dog's feet or anywhere they walked. (Peroxide is a bleaching agent, so test first and/or use with caution on delicate fabrics)
6. Keep an eye on your dog for the next few hours. Keep them calm (confined, if need be) and discourage them from licking the nail. Avoid walking them on rough surfaces or for long distances until you're sure the nail isn't going to re-open.
7. Wrapping the foot is an option, but be sure you're not cutting off circulation and check frequently to make sure the dog isn't invisibly bleeding into the bandage. A quicked nail will eventually stop bleeding on its own if left alone, but it's always better to get it clotted as soon as possible.
8. If your pet has a bleeding disorder or the nail is so badly quicked that you can't get it to stop bleeding, even with several minutes of constant pressure applied, you should seek veterinary assistance.
So, if bleeding's a possibility, why do groomers trim the nails so close? Ideally, pet nails should not touch the floor when the pet is standing normally. When the nails are overgrown and pressing into the floor while walking around, this can interfere with a dog's natural gait and cause joint and muscle issues over time. That's why we like to go short. In order to get nails as short as possible without bleeding them, we have to encourage the blood vessel -- the quick -- inside the nail to recede. The only way to do that is to cut as close to the quick as possible, and although we prefer not to cut into it, it's always a possibility. It can be tricky to judge exactly where the quick is, particularly in dark-colored nails and on a wiggly dog. The good news is that, quickly and calmly dealt with (particularly if you use a styptic powder with an added numbing agent), a light quicking is not going to cause permanent physical or emotional damage. The bigger risk of pain comes from walking around on overgrown nails for long periods of time, so be sure to have your dog's nails trimmed frequently!