Adopting a racing greyhound here in the U.K. is a suprisingly simple process. I'm in the process of adopting from Sheffield Retired Greyhounds, in Yorkshire, and this is the process as far as I understand it-
Step one - Find some greyhounds.
A good way to find a local sanctuary online is to either look up the Retired Greyhound Trust, who have a directory of their branches on their website, or simply to use your favourite seearch engine and type in "Retired Greyhounds (mylocaltown)." You may have to try a few of your local towns, but you'll get one eventually. Another option is to go to your local greyhound track and ask there - Contrary to popular belief, not all greyhound owners and trainers are evil animal-haters who just want to make their little cash-cows run then throw them in the river with a brick around their neck. The majority of greyhounds in sanctuaries or rehoming kennels (The terminology is pretty loose, and I'm not all that consistent in using it myself) are put there by their trainers and owners. Most tracks will have information on where to adopt retired greyhounds, and once you start looking you can't help but trip over adverts with big, soulful eyes and sad pointy muzzles looking at you and asking you to take them home. The Big Issue and many free newspapers carry them, as well as posters in train stations, bus shelters, random advertising hoardings near the track... You get the picture. Once you've got greyhounds on the mind, you'll see them everywhere.
Step two - Decide the sort of greyhound you need.
You'll probably have a few ideas about the sort of dog you want. Male, female, young, old, happy to be alone or in need of constant cuddles, hyperactive or lazy and so forth. If not, start to think about what you can provide, and what you need. If you don't relish the idea of toilet training (and by the way, most greyhounds aren't really house trained in the classical sense whilst they're racing), look for one that's been rehomed before. If you've got other pets - cats, terriers, lizards or pretty much anything that might pique a greyhound's interest, look into getting a cat-safe or pet-safe dog. Sheffield have a "stunt terrier" who is nearly always on hand to show how badly a new dog will react to a smaller animal, and will advertise him accordingly. Incidentally, be wary of sanctuaries who, even after visiting and beginning to talk about adoption, still describe all of their dogs as angels. They're probably not telling the truth. That aside, most places will put an awful lot of hard work and time into getting their dogs back into peak mental condition and getting them used to socialising with humans. Once you've got a rough idea of the dog, and number of dogs you need, go and introduce yourself at the kennels.
As an aside, not all greyhounds are ex-racers. Some are put up for adoption because of injuries before they reach the track and some show no interest in chasing the lure. It's quite possible to get a "retired" greyhound who's never so much as sniffed at the sand track, and is only a few months old - These are usually the best bet if you want a "cat friendly" hound.
Step three - Introduce yourself at the kennel
Phone in advance or get in contact by email before turning up at a kennel. Greyhounds keep an odd schedule, so you may not be welcome when they're taking their afternoon nap, or at mealtimes. Be prepared to get there early in the morning. The kennel-keeper should be happy to show you around the accommodation, so you can see the dogs in their pens, and to answer questions you have about greyhound keeping. You'll be able to take a dog (or a few of them) for a brief walk, and get to know how each dog you're interested in "handles". Try not to just fall for the first one you see - Trust me, they're all that wonderful, personable and friendly. Do this a few times - Come back week-in and week-out, on hot days, cold days, wet days, days so dry your hair turns to straw, to see if you really want to commit to this dog. The kennel may be able to reserve the dog for you, so you can really spend a while getting to know it. Before you go to the kennels, brush up on what you need to know about keeping the dogs. How often should you walk them? How much do they eat? How much space do they need? The kennel staff will tell you all this sort of thing, and will tailor it down to the individual dog (Morland Blasphemy eats more than Dreadful Symmetry, because he's twice her size, but she likes to sprawl out on a bigger bed) but you don't want to look like an ignorant fool when you show up and think that they thrive on lettuce and need a treadmill in the living room to stay comfortable. At the very least banish the myths that they're hard to look after, need endless walks, and are savage. They're not called the forty-five-mile-an-hour-couch-potato for nothing.
Step four - Adopting
Now that you've picked out your dog, visited it a few times, and learnt all you can possibly absorb about the care and feeding of the ex-racing greyhound you can start adopting.
The kennel will normally ask for an adoption fee or donation of one to two hundred pounds, which covers the dog's immunisations, worming, neutering and the general labour costs of keeping the dog for its time in the kennels. You'll often get, along with the dog, a stack of information, a blanket (they get cold easily), a lead, a muzzle (necessary on walks in public places with unknown stimuli) and occassionally a bag of dog food or the like, to help your dog ease into your home. You'll probably get a home visit from the kennel before you adopt the dog, to check that your house is dog safe and suitable for the individual dog you've chosen. Sometimes, they'll bring the dog with them, and if all goes well the dog can be left there with you, their proud new owner, on the day. Some kennels ask for updates on the greyhound at regular intervals, but others don't. It depends on where you go. Many new owners keep in contact anyway, just out of both gratitude and to help foster the sense of community. Kennels are often a hub for greyhound-related activity in the local area, and arrange group walks, amateur shows, fun races (though they can be controversial sometimes) and the like.
Step five - Enjoy your greyhound!