When the worst happens and one of your degus becomes ill, it's obviously time to go to the vet. If you're not already signed up to a forum like Degu World then it would be a good idea to do so first, and post the symptoms and see what other degu owners have to say about it.
Degu World is also useful when it comes to degu-friendly vets, as they have a list of them here for vets in the UK and USA (plus one in Spain) and if you need one close by you could try posting your location and seeing if anyone else knows of one. If you luck out at this stage, call your local vet and test him on degu knowledge before you take your pride and joy to him...it is VERY important that they don't treat your degu as if he or she is a rat or other rodent, deathly important! The Royal College of Veterinary Surgeons also has a searchable database of vets here which you can also limit to specific types of vet as well, and RCVS accredited vets are that little bit better than an unaccredited vet as they've proven their abilities.
If you're the kind of person who likes to plan ahead, a good idea would be to get insurance for your degus. Not everyone does this of course, and it's just as easy to put aside the money you'd be paying to the insurers and only using it for your degus when it's needed, and you'll probably find you never actually need to use it.
UK Vets
Another option is the RSPCA animal hospital, of which there are many littered all over the UK. The PDSA also provides a good service and both of these companies obviously have VERY good history for animal care. Both the RSPCA and PDSA provide low-cost veterinary treatment for your degu if you're on a low income or unemployed, and a treatment should cost no more than a couple of day's worth of benefits and it is dependant on THEIR costs.
A recommended vet if you are in London is C.J. Hall, they are specialists in exotics and have extensive equipment in their two surgeries. One surgery is at 346 Acton Lane, Acton, W3 8NX and the other is at 15 Temple Sheen Road, East Sheen, SW14 7PY. You should expect to pay a lot more for treatment here, but you're paying for experience.
USA Vets
I am hoping that someone from the USA will make suggestions of similar organisations in the USA.
Dealing with the illnesses
Degutopia has a long list of possible illnesses sorted by body part and with descriptions about each and possible solutions. This wiki will also have articles about each of them as well, with as much important information as possible from people who have had degus with the illness and how they coped. Don't forget you can also post in places like Degu World (forum in the "useful websites") and other degu owners will give advice.